Permutation groups¶
A permutation group is a finite group \(G\) whose elements are permutations of a given finite set \(X\) (i.e., bijections \(X \longrightarrow X\)) and whose group operation is the composition of permutations. The number of elements of \(X\) is called the degree of \(G\).
In Sage, a permutation is represented as either a string that defines a permutation using disjoint cycle notation, or a list of tuples, which represent disjoint cycles. That is:
(a,...,b)(c,...,d)...(e,...,f)  <--> [(a,...,b), (c,...,d),..., (e,...,f)]
                  () = identity <--> []
You can make the “named” permutation groups (see
permgp_named.py) and use the following
constructions:
- permutation group generated by elements, 
- direct_product_permgroups, which takes a list of permutation groups and returns their direct product.
JOKE: Q: What’s hot, chunky, and acts on a polygon? A: Dihedral soup. Renteln, P. and Dundes, A. “Foolproof: A Sampling of Mathematical Folk Humor.” Notices Amer. Math. Soc. 52, 24-34, 2005.
Index of methods¶
Here are the methods of a PermutationGroup()
| Return whether  | |
| Return a finitely presented group isomorphic to  | |
| Return the list of block systems of imprimitivity. | |
| Return the number of elements of this group. | |
| Return the subgroup of elements that commute with every element of this group. | |
| Return the centralizer of  | |
| Return a group character from  | |
| Return the matrix of values of the irreducible characters of a permutation group \(G\) at the conjugacy classes of \(G\). | |
| Compute the group cohomology \(H^n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. | |
| Compute the p-part of the group cohomology \(H^n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. | |
| Return the commutator subgroup of a group, or of a pair of groups. | |
| Return the composition series of this group as a list of permutation groups. | |
| Return the conjugacy class of  | |
| Return a list with all the conjugacy classes of  | |
| Return a complete list of representatives of conjugacy classes in a permutation group \(G\). | |
| Return a complete list of representatives of conjugacy classes of subgroups in a permutation group \(G\). | |
| Return the group formed by conjugating  | |
| Return the construction of  | |
| Return a list of the cosets of  | |
| Return the degree of this permutation group. | |
| Return the derived series of this group as a list of permutation groups. | |
| Wraps GAP’s  | |
| Return the underlying set that this permutation group acts on. | |
| Compute the exponent of the group. | |
| Return the Fitting subgroup of  | |
| Return the list of points fixed by  | |
| Return the Frattini subgroup of  | |
| Return the \(i\)-th generator of  | |
| Return tuple of generators of this group. | |
| For this group, returns a generating set which has few elements. | |
| Return the ID code of this group, which is a list of two integers. | |
| Return the index of this group in the GAP database of primitive groups. | |
| The holomorph of a group as a permutation group. | |
| Compute the group homology \(H_n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. Wraps HAP’s  | |
| Compute the \(p\)-part of the group homology \(H_n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. Wraps HAP’s  | |
| Return the ID code of this group, which is a list of two integers. | |
| Return the permutation group that is the intersection of  | |
| Return a list of the irreducible characters of  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return  | |
| Return an isomorphism from  | |
| If the group is simple, then this returns the name of the group. | |
| Return an iterator over the elements of this group. | |
| Return the largest point moved by a permutation in this group. | |
| Return list of all elements of this group. | |
| Return the lower central series of this group as a list of permutation groups. | |
| Return the maximal proper normal subgroups of  | |
| Return a minimal generating set. | |
| Return the nontrivial minimal normal subgroups  | |
| Return the Molien series of a permutation group. The function | |
| Return the number of generators of  | |
| Return the list of points not fixed by  | |
| Return the normal subgroups of this group as a (sorted in increasing order) list of permutation groups. | |
| Return the normalizer of  | |
| Return  | |
| Return the Poincaré series of \(G \mod p\) (\(p \geq 2\) must be a prime), for \(n\) large. | |
| Return a random element of this group. | |
| Return an element of  | |
| The semidirect product of  | |
| Return the sign representation of  | |
| Return the socle of  | |
| Return the solvable radical of  | |
| Return the subgroup of  | |
| Return a Strong Generating System of  | |
| Return a string that tries to describe the structure of  | |
| Wraps the  | |
| Return a list of all the subgroups of  | |
| Return a Sylow \(p\)-subgroup of the finite group \(G\), where \(p\) is a prime. | |
| If G is a permutation group acting on the set \(X = \{1, 2, ...., n\}\) and H is the stabilizer subgroup of <integer>, a right (respectively left) transversal is a set containing exactly one element from each right (respectively left) coset of H. This method returns a right transversal of  | |
| Return the trivial character of  | |
| Return the upper central series of this group as a list of permutation groups. | 
AUTHORS:
- David Joyner (2005-10-14): first version 
- David Joyner (2005-11-17) 
- William Stein (2005-11-26): rewrite to better wrap Gap 
- David Joyner (2005-12-21) 
- William Stein and David Joyner (2006-01-04): added conjugacy_class_representatives 
- David Joyner (2006-03): reorganization into subdirectory perm_gps; added __contains__, has_element; fixed _cmp_; added subgroup class+methods, PGL,PSL,PSp, PSU classes, 
- David Joyner (2006-06): added PGU, functionality to SymmetricGroup, AlternatingGroup, direct_product_permgroups 
- David Joyner (2006-08): added degree, ramification_module_decomposition_modular_curve and ramification_module_decomposition_hurwitz_curve methods to PSL(2,q), MathieuGroup, is_isomorphic 
- Bobby Moretti (2006)-10): Added KleinFourGroup, fixed bug in DihedralGroup 
- David Joyner (2006-10): added is_subgroup (fixing a bug found by Kiran Kedlaya), is_solvable, normalizer, is_normal_subgroup, Suzuki 
- David Kohel (2007-02): fixed __contains__ to not enumerate group elements, following the convention for __call__ 
- David Harvey, Mike Hansen, Nick Alexander, William Stein (2007-02,03,04,05): Various patches 
- Nathan Dunfield (2007-05): added orbits 
- David Joyner (2007-06): added subgroup method (suggested by David Kohel), composition_series, lower_central_series, upper_central_series, cayley_table, quotient_group, sylow_subgroup, is_cyclic, homology, homology_part, cohomology, cohomology_part, poincare_series, molien_series, is_simple, is_monomial, is_supersolvable, is_nilpotent, is_perfect, is_polycyclic, is_elementary_abelian, is_pgroup, gens_small, isomorphism_type_info_simple_group. moved all the”named” groups to a new file. 
- Nick Alexander (2007-07): move is_isomorphic to isomorphism_to, add from_gap_list 
- William Stein (2007-07): put is_isomorphic back (and make it better) 
- David Joyner (2007-08): fixed bugs in composition_series, upper/lower_central_series, derived_series, 
- David Joyner (2008-06): modified is_normal (reported by W. J. Palenstijn), and added normalizes 
- David Joyner (2008-08): Added example to docstring of cohomology. 
- Simon King (2009-04): __cmp__ methods for PermutationGroup_generic and PermutationGroup_subgroup 
- Nicolas Borie (2009): Added orbit, transversals, stabiliser and strong_generating_system methods 
- Christopher Swenson (2012): Added a special case to compute the order efficiently. (This patch Copyright 2012 Google Inc. All Rights Reserved. ) 
- Javier Lopez Pena (2013): Added conjugacy classes. 
- Sebastian Oehms (2018): added _coerce_map_from_ in order to use isomorphism coming up with as_permutation_group method (Issue #25706) 
- Christian Stump (2018): Added alternative implementation of strong_generating_system directly using GAP. 
- Sebastian Oehms (2018): Added - PermutationGroup_generic._Hom_()to use- sage.groups.libgap_morphism.GroupHomset_libgapand- PermutationGroup_generic.gap()and- PermutationGroup_generic._subgroup_constructor()(for compatibility to libgap framework, see Issue #26750
REFERENCES:
- Cameron, P., Permutation Groups. New York: Cambridge University Press, 1999. 
- Wielandt, H., Finite Permutation Groups. New York: Academic Press, 1964. 
- Dixon, J. and Mortimer, B., Permutation Groups, Springer-Verlag, Berlin/New York, 1996. 
Note
Though Suzuki groups are okay, Ree groups should not be wrapped as permutation groups - the construction is too slow - unless (for small values or the parameter) they are made using explicit generators.
- sage.groups.perm_gps.permgroup.PermutationGroup(gens=None, *args, **kwds)[source]¶
- Return the permutation group associated to \(x\) (typically a list of generators). - INPUT: - gens– (default:- None) list of generators
- gap_group– (optional) a gap permutation group
- canonicalize– boolean (default:- True); if- True, sort generators and remove duplicates
 - OUTPUT: a permutation group - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: G Permutation Group with generators [(3,4), (1,2,3)(4,5)] - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]]) >>> G Permutation Group with generators [(3,4), (1,2,3)(4,5)] - We can also make permutation groups from PARI groups: - sage: # needs sage.libs.pari sage: H = pari('x^4 - 2*x^3 - 2*x + 1').polgalois() sage: G = PariGroup(H, 4); G PARI group [8, -1, 3, "D(4)"] of degree 4 sage: H = PermutationGroup(G); H Transitive group number 3 of degree 4 sage: H.gens() ((1,2,3,4), (1,3)) - >>> from sage.all import * >>> # needs sage.libs.pari >>> H = pari('x^4 - 2*x^3 - 2*x + 1').polgalois() >>> G = PariGroup(H, Integer(4)); G PARI group [8, -1, 3, "D(4)"] of degree 4 >>> H = PermutationGroup(G); H Transitive group number 3 of degree 4 >>> H.gens() ((1,2,3,4), (1,3)) - We can also create permutation groups whose generators are GAP permutation objects: - sage: p = gap('(1,2)(3,7)(4,6)(5,8)'); p (1,2)(3,7)(4,6)(5,8) sage: PermutationGroup([p]) Permutation Group with generators [(1,2)(3,7)(4,6)(5,8)] - >>> from sage.all import * >>> p = gap('(1,2)(3,7)(4,6)(5,8)'); p (1,2)(3,7)(4,6)(5,8) >>> PermutationGroup([p]) Permutation Group with generators [(1,2)(3,7)(4,6)(5,8)] - Permutation groups can work on any domain. In the following examples, the permutations are specified in list notation, according to the order of the elements of the domain: - sage: list(PermutationGroup([['b','c','a']], domain=['a','b','c'])) [(), ('a','b','c'), ('a','c','b')] sage: list(PermutationGroup([['b','c','a']], domain=['b','c','a'])) [()] sage: list(PermutationGroup([['b','c','a']], domain=['a','c','b'])) [(), ('a','b')] - >>> from sage.all import * >>> list(PermutationGroup([['b','c','a']], domain=['a','b','c'])) [(), ('a','b','c'), ('a','c','b')] >>> list(PermutationGroup([['b','c','a']], domain=['b','c','a'])) [()] >>> list(PermutationGroup([['b','c','a']], domain=['a','c','b'])) [(), ('a','b')] - There is an underlying gap object that implements each permutation group: - sage: G = PermutationGroup([[(1,2,3,4)]]) sage: G._gap_() Group( [ (1,2,3,4) ] ) sage: gap(G) Group( [ (1,2,3,4) ] ) sage: gap(G) is G._gap_() True sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: current_randstate().set_seed_gap() sage: G1, G2 = G._gap_().DerivedSeries() sage: G1 Group( [ (3,4), (1,2,3)(4,5) ] ) sage: G2.GeneratorsSmallest() [ (3,4,5), (2,3)(4,5), (1,2)(4,5) ] - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> G._gap_() Group( [ (1,2,3,4) ] ) >>> gap(G) Group( [ (1,2,3,4) ] ) >>> gap(G) is G._gap_() True >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]]) >>> current_randstate().set_seed_gap() >>> G1, G2 = G._gap_().DerivedSeries() >>> G1 Group( [ (3,4), (1,2,3)(4,5) ] ) >>> G2.GeneratorsSmallest() [ (3,4,5), (2,3)(4,5), (1,2)(4,5) ] - We can create a permutation group from a group action: - sage: a = lambda x: (2*x) % 7 sage: H = PermutationGroup(action=a, domain=range(7)) # needs sage.combinat sage: H.orbits() # needs sage.libs.pari ((0,), (1, 2, 4), (3, 6, 5)) sage: H.gens() # needs sage.libs.pari ((1,2,4), (3,6,5)) - >>> from sage.all import * >>> a = lambda x: (Integer(2)*x) % Integer(7) >>> H = PermutationGroup(action=a, domain=range(Integer(7))) # needs sage.combinat >>> H.orbits() # needs sage.libs.pari ((0,), (1, 2, 4), (3, 6, 5)) >>> H.gens() # needs sage.libs.pari ((1,2,4), (3,6,5)) - Note that we provide generators for the acting group. The permutation group we construct is its homomorphic image: - sage: # needs sage.combinat sage: a = lambda g, x: vector(g*x, immutable=True) sage: X = [vector(x, immutable=True) for x in GF(3)^2] sage: G = SL(2,3); G.gens() ( [1 1] [0 1] [0 1], [2 0] ) sage: H = PermutationGroup(G.gens(), action=a, domain=X) sage: H.orbits() (((0, 0),), ((1, 0), (2, 0), (0, 1), (1, 1), (2, 1), (0, 2), (1, 2), (2, 2))) sage: H.gens() (((0,1),(1,1),(2,1))((0,2),(2,2),(1,2)), ((1,0),(0,2),(2,0),(0,1))((1,1),(1,2),(2,2),(2,1))) - >>> from sage.all import * >>> # needs sage.combinat >>> a = lambda g, x: vector(g*x, immutable=True) >>> X = [vector(x, immutable=True) for x in GF(Integer(3))**Integer(2)] >>> G = SL(Integer(2),Integer(3)); G.gens() ( [1 1] [0 1] [0 1], [2 0] ) >>> H = PermutationGroup(G.gens(), action=a, domain=X) >>> H.orbits() (((0, 0),), ((1, 0), (2, 0), (0, 1), (1, 1), (2, 1), (0, 2), (1, 2), (2, 2))) >>> H.gens() (((0,1),(1,1),(2,1))((0,2),(2,2),(1,2)), ((1,0),(0,2),(2,0),(0,1))((1,1),(1,2),(2,2),(2,1))) - The orbits of the conjugation action are the conjugacy classes, i.e., in bijection with integer partitions: - sage: a = lambda g, x: g*x*g^-1 sage: [len(PermutationGroup(SymmetricGroup(n).gens(), action=a, # needs sage.combinat ....: domain=SymmetricGroup(n)).orbits()) ....: for n in range(1, 8)] [1, 2, 3, 5, 7, 11, 15] - >>> from sage.all import * >>> a = lambda g, x: g*x*g**-Integer(1) >>> [len(PermutationGroup(SymmetricGroup(n).gens(), action=a, # needs sage.combinat ... domain=SymmetricGroup(n)).orbits()) ... for n in range(Integer(1), Integer(8))] [1, 2, 3, 5, 7, 11, 15] 
- class sage.groups.perm_gps.permgroup.PermutationGroup_action(gens, action, domain, gap_group=None, category=None, canonicalize=None)[source]¶
- Bases: - PermutationGroup_generic- A permutation group given by a finite group action. - EXAMPLES: - A cyclic action: - sage: n = 3 sage: a = lambda x: SetPartition([[e % n + 1 for e in b] for b in x]) sage: S = SetPartitions(n) # needs sage.combinat sage: G = PermutationGroup(action=a, domain=S) # needs sage.combinat sage: G.orbits() # needs sage.combinat (({{1}, {2}, {3}},), ({{1, 2}, {3}}, {{1}, {2, 3}}, {{1, 3}, {2}}), ({{1, 2, 3}},)) - >>> from sage.all import * >>> n = Integer(3) >>> a = lambda x: SetPartition([[e % n + Integer(1) for e in b] for b in x]) >>> S = SetPartitions(n) # needs sage.combinat >>> G = PermutationGroup(action=a, domain=S) # needs sage.combinat >>> G.orbits() # needs sage.combinat (({{1}, {2}, {3}},), ({{1, 2}, {3}}, {{1}, {2, 3}}, {{1, 3}, {2}}), ({{1, 2, 3}},)) - The regular action of the symmetric group: - sage: a = lambda g, x: g*x*g^-1 sage: S = SymmetricGroup(3) sage: G = PermutationGroup(S.gens(), action=a, domain=S) # needs sage.combinat sage: G.orbits() # needs sage.combinat (((),), ((1,3,2), (1,2,3)), ((2,3), (1,3), (1,2))) - >>> from sage.all import * >>> a = lambda g, x: g*x*g**-Integer(1) >>> S = SymmetricGroup(Integer(3)) >>> G = PermutationGroup(S.gens(), action=a, domain=S) # needs sage.combinat >>> G.orbits() # needs sage.combinat (((),), ((1,3,2), (1,2,3)), ((2,3), (1,3), (1,2))) - The trivial action of the symmetric group: - sage: PermutationGroup(SymmetricGroup(3).gens(), # needs sage.combinat ....: action=lambda g, x: x, domain=[1]) Permutation Group with generators [()] - >>> from sage.all import * >>> PermutationGroup(SymmetricGroup(Integer(3)).gens(), # needs sage.combinat ... action=lambda g, x: x, domain=[Integer(1)]) Permutation Group with generators [()] - orbits()[source]¶
- Return the orbits of the elements of the domain under the default group action. - EXAMPLES: - sage: a = lambda x: (2*x) % 7 sage: G = PermutationGroup(action=a, domain=range(7)) # needs sage.combinat sage: G.orbits() # needs sage.combinat ((0,), (1, 2, 4), (3, 6, 5)) - >>> from sage.all import * >>> a = lambda x: (Integer(2)*x) % Integer(7) >>> G = PermutationGroup(action=a, domain=range(Integer(7))) # needs sage.combinat >>> G.orbits() # needs sage.combinat ((0,), (1, 2, 4), (3, 6, 5)) 
 
- class sage.groups.perm_gps.permgroup.PermutationGroup_generic(gens=None, gap_group=None, canonicalize=True, domain=None, category=None)[source]¶
- Bases: - FiniteGroup- A generic permutation group. - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: G Permutation Group with generators [(3,4), (1,2,3)(4,5)] sage: G.center() Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) sage: G.group_id() [120, 34] sage: n = G.order(); n 120 sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: TestSuite(G).run() - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]]) >>> G Permutation Group with generators [(3,4), (1,2,3)(4,5)] >>> G.center() Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]) >>> G.group_id() [120, 34] >>> n = G.order(); n 120 >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]]) >>> TestSuite(G).run() - Element[source]¶
- alias of - PermutationGroupElement
 - Subgroup[source]¶
- alias of - PermutationGroup_subgroup
 - are_conjugate(H1, H2)[source]¶
- Return whether - H1and- H2are conjugate subgroups in- G.- EXAMPLES: - sage: G = SymmetricGroup(3) sage: H1 = PermutationGroup([(1,2)]) sage: H2 = PermutationGroup([(2,3)]) sage: G.are_conjugate(H1, H2) True sage: G = SymmetricGroup(4) sage: H1 = PermutationGroup([[(1,3),(2,4)], [(1,2),(3,4)]]) sage: H2 = PermutationGroup([[(1,2)], [(1,2),(3,4)]]) sage: G.are_conjugate(H1, H2) False - >>> from sage.all import * >>> G = SymmetricGroup(Integer(3)) >>> H1 = PermutationGroup([(Integer(1),Integer(2))]) >>> H2 = PermutationGroup([(Integer(2),Integer(3))]) >>> G.are_conjugate(H1, H2) True >>> G = SymmetricGroup(Integer(4)) >>> H1 = PermutationGroup([[(Integer(1),Integer(3)),(Integer(2),Integer(4))], [(Integer(1),Integer(2)),(Integer(3),Integer(4))]]) >>> H2 = PermutationGroup([[(Integer(1),Integer(2))], [(Integer(1),Integer(2)),(Integer(3),Integer(4))]]) >>> G.are_conjugate(H1, H2) False 
 - as_finitely_presented_group(reduced=False)[source]¶
- Return a finitely presented group isomorphic to - self.- This method acts as wrapper for the GAP function - IsomorphismFpGroupByGenerators, which yields an isomorphism from a given group to a finitely presented group.- INPUT: - reduced– (default:- False) if- True,- FinitelyPresentedGroup.simplifiedis called, attempting to simplify the presentation of the finitely presented group to be returned
 - OUTPUT: finite presentation of - self, obtained by taking the image of the isomorphism returned by the GAP function- IsomorphismFpGroupByGenerators- ALGORITHM: uses GAP - EXAMPLES: - sage: CyclicPermutationGroup(50).as_finitely_presented_group() Finitely presented group < a | a^50 > sage: DihedralGroup(4).as_finitely_presented_group() Finitely presented group < a, b | b^2, a^4, (b*a)^2 > sage: GeneralDihedralGroup([2,2]).as_finitely_presented_group() Finitely presented group < a, b, c | a^2, b^2, c^2, (c*b)^2, (c*a)^2, (b*a)^2 > - >>> from sage.all import * >>> CyclicPermutationGroup(Integer(50)).as_finitely_presented_group() Finitely presented group < a | a^50 > >>> DihedralGroup(Integer(4)).as_finitely_presented_group() Finitely presented group < a, b | b^2, a^4, (b*a)^2 > >>> GeneralDihedralGroup([Integer(2),Integer(2)]).as_finitely_presented_group() Finitely presented group < a, b, c | a^2, b^2, c^2, (c*b)^2, (c*a)^2, (b*a)^2 > - GAP algorithm is not guaranteed to produce minimal or canonical presentation: - sage: G = PermutationGroup(['(1,2,3,4,5)', '(1,5)(2,4)']) sage: G.is_isomorphic(DihedralGroup(5)) True sage: K = G.as_finitely_presented_group(); K Finitely presented group < a, b | b^2, (b*a)^2, b*a^-3*b*a^2 > sage: K.as_permutation_group().is_isomorphic(DihedralGroup(5)) True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3,4,5)', '(1,5)(2,4)']) >>> G.is_isomorphic(DihedralGroup(Integer(5))) True >>> K = G.as_finitely_presented_group(); K Finitely presented group < a, b | b^2, (b*a)^2, b*a^-3*b*a^2 > >>> K.as_permutation_group().is_isomorphic(DihedralGroup(Integer(5))) True - We can attempt to reduce the output presentation: - sage: H = PermutationGroup(['(1,2,3,4,5)', '(1,3,5,2,4)']) sage: H.as_finitely_presented_group() Finitely presented group < a, b | b^-2*a^-1, b*a^-2 > sage: H.as_finitely_presented_group(reduced=True) Finitely presented group < a | a^5 > - >>> from sage.all import * >>> H = PermutationGroup(['(1,2,3,4,5)', '(1,3,5,2,4)']) >>> H.as_finitely_presented_group() Finitely presented group < a, b | b^-2*a^-1, b*a^-2 > >>> H.as_finitely_presented_group(reduced=True) Finitely presented group < a | a^5 > - AUTHORS: - Davis Shurbert (2013-06-21): initial version 
 
 - base(seed=None)[source]¶
- Return a (minimum) base of this permutation group. - A base \(B\) of a permutation group is a subset of the domain of the group such that the only group element stabilizing all of \(B\) is the identity. - INPUT: - seed– (default:- None) if given must be a subset of the domain of a base. When used, an attempt to create a base containing all or part of- seedwill be made.
 - EXAMPLES: - sage: G = PermutationGroup([(1,2,3),(6,7,8)]) sage: G.base() [1, 6] sage: G.base([2]) [2, 6] sage: H = PermutationGroup([('a','b','c'),('a','y')]) sage: H.base() ['a', 'b', 'c'] sage: S = SymmetricGroup(13) sage: S.base() [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] sage: S = MathieuGroup(12) sage: S.base() [1, 2, 3, 4, 5] sage: S.base([1,3,5,7,9,11]) # create a base for M12 with only odd integers [1, 3, 5, 7, 9] - >>> from sage.all import * >>> G = PermutationGroup([(Integer(1),Integer(2),Integer(3)),(Integer(6),Integer(7),Integer(8))]) >>> G.base() [1, 6] >>> G.base([Integer(2)]) [2, 6] >>> H = PermutationGroup([('a','b','c'),('a','y')]) >>> H.base() ['a', 'b', 'c'] >>> S = SymmetricGroup(Integer(13)) >>> S.base() [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] >>> S = MathieuGroup(Integer(12)) >>> S.base() [1, 2, 3, 4, 5] >>> S.base([Integer(1),Integer(3),Integer(5),Integer(7),Integer(9),Integer(11)]) # create a base for M12 with only odd integers [1, 3, 5, 7, 9] 
 - blocks_all(representatives=True)[source]¶
- Return the list of block systems of imprimitivity. - For more information on primitivity, see the Wikipedia article on primitive group actions. - INPUT: - representative– boolean; whether to return all possible block systems of imprimitivity or only one of their representatives (the block can be obtained from its representative set \(S\) by computing the orbit of \(S\) under- self).- This parameter is set to - Trueby default (as it is GAP’s default behaviour).
 - OUTPUT: - This method returns a description of all block systems. Hence, the output is a “list of lists of lists” or a “list of lists” depending on the value of - representatives. A bit more clearly, output is:- A list of length (#number of different block systems) of - block systems, each of them being defined as - If - representatives=True: a list of representatives of each set of the block system
- If - representatives=False: a partition of the elements defining an imprimitivity block.
 
 
 - See also - EXAMPLES: - Picking an interesting group: - sage: # needs sage.graphs sage: g = graphs.DodecahedralGraph() sage: g.is_vertex_transitive() True sage: ag = g.automorphism_group() sage: ag.is_primitive() False - >>> from sage.all import * >>> # needs sage.graphs >>> g = graphs.DodecahedralGraph() >>> g.is_vertex_transitive() True >>> ag = g.automorphism_group() >>> ag.is_primitive() False - Computing its blocks representatives: - sage: ag.blocks_all() # needs sage.graphs [[0, 15]] - >>> from sage.all import * >>> ag.blocks_all() # needs sage.graphs [[0, 15]] - Now the full block: - sage: sorted(ag.blocks_all(representatives=False)[0]) # needs sage.graphs [[0, 15], [1, 16], [2, 12], [3, 13], [4, 9], [5, 10], [6, 11], [7, 18], [8, 17], [14, 19]] - >>> from sage.all import * >>> sorted(ag.blocks_all(representatives=False)[Integer(0)]) # needs sage.graphs [[0, 15], [1, 16], [2, 12], [3, 13], [4, 9], [5, 10], [6, 11], [7, 18], [8, 17], [14, 19]] 
 - cardinality()[source]¶
- Return the number of elements of this group. - See also: - degree().- EXAMPLES: - sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]]) sage: G.order() 12 sage: G = PermutationGroup([()]) sage: G.order() 1 sage: G = PermutationGroup([]) sage: G.order() 1 - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))], [(Integer(1),Integer(2))]]) >>> G.order() 12 >>> G = PermutationGroup([()]) >>> G.order() 1 >>> G = PermutationGroup([]) >>> G.order() 1 - cardinality()is just an alias:- sage: PermutationGroup([(1,2,3)]).cardinality() 3 - >>> from sage.all import * >>> PermutationGroup([(Integer(1),Integer(2),Integer(3))]).cardinality() 3 
 - center()[source]¶
- Return the subgroup of elements that commute with every element of this group. - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3,4)]]) sage: G.center() Subgroup generated by [(1,2,3,4)] of (Permutation Group with generators [(1,2,3,4)]) sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]]) sage: G.center() Subgroup generated by [()] of (Permutation Group with generators [(1,2), (1,2,3,4)]) - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> G.center() Subgroup generated by [(1,2,3,4)] of (Permutation Group with generators [(1,2,3,4)]) >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))], [(Integer(1),Integer(2))]]) >>> G.center() Subgroup generated by [()] of (Permutation Group with generators [(1,2), (1,2,3,4)]) 
 - centralizer(g)[source]¶
- Return the centralizer of - gin- self.- EXAMPLES: - sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) sage: g = G([(1,3)]) sage: G.centralizer(g) Subgroup generated by [(1,3), (2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) sage: g = G([(1,2,3,4)]) sage: G.centralizer(g) Subgroup generated by [(1,2,3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) sage: H = G.subgroup([G([(1,2,3,4)])]) sage: G.centralizer(H) Subgroup generated by [(1,2,3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> g = G([(Integer(1),Integer(3))]) >>> G.centralizer(g) Subgroup generated by [(1,3), (2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) >>> g = G([(Integer(1),Integer(2),Integer(3),Integer(4))]) >>> G.centralizer(g) Subgroup generated by [(1,2,3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) >>> H = G.subgroup([G([(Integer(1),Integer(2),Integer(3),Integer(4))])]) >>> G.centralizer(H) Subgroup generated by [(1,2,3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) 
 - character(values)[source]¶
- Return a group character from - values, where- valuesis a list of the values of the character evaluated on the conjugacy classes.- EXAMPLES: - sage: G = AlternatingGroup(4) sage: n = len(G.conjugacy_classes_representatives()) sage: G.character([1]*n) # needs sage.rings.number_field Character of Alternating group of order 4!/2 as a permutation group - >>> from sage.all import * >>> G = AlternatingGroup(Integer(4)) >>> n = len(G.conjugacy_classes_representatives()) >>> G.character([Integer(1)]*n) # needs sage.rings.number_field Character of Alternating group of order 4!/2 as a permutation group 
 - character_table()[source]¶
- Return the matrix of values of the irreducible characters of a permutation group \(G\) at the conjugacy classes of \(G\). - The columns represent the conjugacy classes of \(G\) and the rows represent the different irreducible characters in the ordering given by GAP. - EXAMPLES: - sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]]) sage: G.order() 12 sage: G.character_table() # needs sage.rings.number_field [ 1 1 1 1] [ 1 -zeta3 - 1 zeta3 1] [ 1 zeta3 -zeta3 - 1 1] [ 3 0 0 -1] sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]]) sage: CT = gap(G).CharacterTable() - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3))]]) >>> G.order() 12 >>> G.character_table() # needs sage.rings.number_field [ 1 1 1 1] [ 1 -zeta3 - 1 zeta3 1] [ 1 zeta3 -zeta3 - 1 1] [ 3 0 0 -1] >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3))]]) >>> CT = gap(G).CharacterTable() - Type - CT.Display()to display this nicely.- sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) sage: G.order() 8 sage: G.character_table() # needs sage.rings.number_field [ 1 1 1 1 1] [ 1 -1 -1 1 1] [ 1 -1 1 -1 1] [ 1 1 -1 -1 1] [ 2 0 0 0 -2] sage: CT = gap(G).CharacterTable() - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> G.order() 8 >>> G.character_table() # needs sage.rings.number_field [ 1 1 1 1 1] [ 1 -1 -1 1 1] [ 1 -1 1 -1 1] [ 1 1 -1 -1 1] [ 2 0 0 0 -2] >>> CT = gap(G).CharacterTable() - Again, type - CT.Display()to display this nicely.- sage: # needs sage.rings.number_field sage: SymmetricGroup(2).character_table() [ 1 -1] [ 1 1] sage: SymmetricGroup(3).character_table() [ 1 -1 1] [ 2 0 -1] [ 1 1 1] sage: SymmetricGroup(5).character_table() [ 1 -1 1 1 -1 -1 1] [ 4 -2 0 1 1 0 -1] [ 5 -1 1 -1 -1 1 0] [ 6 0 -2 0 0 0 1] [ 5 1 1 -1 1 -1 0] [ 4 2 0 1 -1 0 -1] [ 1 1 1 1 1 1 1] sage: list(AlternatingGroup(6).character_table()) [(1, 1, 1, 1, 1, 1, 1), (5, 1, 2, -1, -1, 0, 0), (5, 1, -1, 2, -1, 0, 0), (8, 0, -1, -1, 0, zeta5^3 + zeta5^2 + 1, -zeta5^3 - zeta5^2), (8, 0, -1, -1, 0, -zeta5^3 - zeta5^2, zeta5^3 + zeta5^2 + 1), (9, 1, 0, 0, 1, -1, -1), (10, -2, 1, 1, 0, 0, 0)] - >>> from sage.all import * >>> # needs sage.rings.number_field >>> SymmetricGroup(Integer(2)).character_table() [ 1 -1] [ 1 1] >>> SymmetricGroup(Integer(3)).character_table() [ 1 -1 1] [ 2 0 -1] [ 1 1 1] >>> SymmetricGroup(Integer(5)).character_table() [ 1 -1 1 1 -1 -1 1] [ 4 -2 0 1 1 0 -1] [ 5 -1 1 -1 -1 1 0] [ 6 0 -2 0 0 0 1] [ 5 1 1 -1 1 -1 0] [ 4 2 0 1 -1 0 -1] [ 1 1 1 1 1 1 1] >>> list(AlternatingGroup(Integer(6)).character_table()) [(1, 1, 1, 1, 1, 1, 1), (5, 1, 2, -1, -1, 0, 0), (5, 1, -1, 2, -1, 0, 0), (8, 0, -1, -1, 0, zeta5^3 + zeta5^2 + 1, -zeta5^3 - zeta5^2), (8, 0, -1, -1, 0, -zeta5^3 - zeta5^2, zeta5^3 + zeta5^2 + 1), (9, 1, 0, 0, 1, -1, -1), (10, -2, 1, 1, 0, 0, 0)] - Suppose that you have a class function \(f(g)\) on \(G\) and you know the values \(v_1, \ldots, v_n\) on the conjugacy class elements in - conjugacy_classes_representatives(G)= \([g_1, \ldots, g_n]\). Since the irreducible characters \(\rho_1, \ldots, \rho_n\) of \(G\) form an \(E\)-basis of the space of all class functions (\(E\) a “sufficiently large” cyclotomic field), such a class function is a linear combination of these basis elements, \(f = c_1 \rho_1 + \cdots + c_n \rho_n\). To find the coefficients \(c_i\), you simply solve the linear system- character_table_values(G)\([v_1, ..., v_n] = [c_1, ..., c_n]\), where \([v_1, \ldots, v_n]\) =- character_table_values(G)\(^{(-1)}[c_1, ..., c_n]\).- AUTHORS: - David Joyner and William Stein (2006-01-04) 
 
 - cohomology(n, p=0)[source]¶
- Compute the group cohomology \(H^n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. - Wraps HAP’s - GroupHomologyfunction, written by Graham Ellis.- REQUIRES: GAP package HAP (in gap_packages-*.spkg). - EXAMPLES: - sage: G = SymmetricGroup(4) sage: G.cohomology(1,2) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 sage: G = SymmetricGroup(3) sage: G.cohomology(5) # optional - gap_package_hap Trivial Abelian group sage: G.cohomology(5,2) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 sage: G.homology(5,3) # optional - gap_package_hap Trivial Abelian group sage: G.homology(5,4) # optional - gap_package_hap Traceback (most recent call last): ... ValueError: p must be 0 or prime - >>> from sage.all import * >>> G = SymmetricGroup(Integer(4)) >>> G.cohomology(Integer(1),Integer(2)) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 >>> G = SymmetricGroup(Integer(3)) >>> G.cohomology(Integer(5)) # optional - gap_package_hap Trivial Abelian group >>> G.cohomology(Integer(5),Integer(2)) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 >>> G.homology(Integer(5),Integer(3)) # optional - gap_package_hap Trivial Abelian group >>> G.homology(Integer(5),Integer(4)) # optional - gap_package_hap Traceback (most recent call last): ... ValueError: p must be 0 or prime - This computes \(H^4(S_3, \ZZ)\) and \(H^4(S_3, \ZZ / 2 \ZZ)\), respectively. - AUTHORS: - David Joyner and Graham Ellis 
 - REFERENCES: - G. Ellis, ‘Computing group resolutions’, J. Symbolic Computation. Vol.38, (2004)1077-1118 (Available at http://hamilton.nuigalway.ie/). 
- D. Joyner, ‘A primer on computational group homology and cohomology’, http://front.math.ucdavis.edu/0706.0549. 
 
 - cohomology_part(n, p=0)[source]¶
- Compute the p-part of the group cohomology \(H^n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. - Wraps HAP’s Homology function, written by Graham Ellis, applied to the \(p\)-Sylow subgroup of \(G\). - REQUIRES: GAP package HAP (in gap_packages-*.spkg). - EXAMPLES: - sage: G = SymmetricGroup(5) sage: G.cohomology_part(7,2) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 x C2 x C2 sage: G = SymmetricGroup(3) sage: G.cohomology_part(2,3) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C3 - >>> from sage.all import * >>> G = SymmetricGroup(Integer(5)) >>> G.cohomology_part(Integer(7),Integer(2)) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 x C2 x C2 >>> G = SymmetricGroup(Integer(3)) >>> G.cohomology_part(Integer(2),Integer(3)) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C3 - AUTHORS: - David Joyner and Graham Ellis 
 
 - commutator(other=None)[source]¶
- Return the commutator subgroup of a group, or of a pair of groups. - INPUT: - other– (default:- None) a permutation group
 - OUTPUT: - Let \(G\) denote - self. If- otheris- None, then this method returns the subgroup of \(G\) generated by the set of commutators,\[\{[g_1,g_2]\vert g_1, g_2\in G\} = \{g_1^{-1}g_2^{-1}g_1g_2\vert g_1, g_2\in G\}\]- Let \(H\) denote - other, in the case that it is not- None. Then this method returns the group generated by the set of commutators,\[\{[g,h]\vert g\in G\, h\in H\} = \{g^{-1}h^{-1}gh\vert g\in G\, h\in H\}\]- The two groups need only be permutation groups, there is no notion of requiring them to explicitly be subgroups of some other group. - Note - For the identical statement, the generators of the returned group can vary from one execution to the next. - EXAMPLES: - sage: G = DiCyclicGroup(4) sage: G.commutator() Permutation Group with generators [(1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)] sage: G = SymmetricGroup(5) sage: H = CyclicPermutationGroup(5) sage: C = G.commutator(H) sage: C.is_isomorphic(AlternatingGroup(5)) True - >>> from sage.all import * >>> G = DiCyclicGroup(Integer(4)) >>> G.commutator() Permutation Group with generators [(1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)] >>> G = SymmetricGroup(Integer(5)) >>> H = CyclicPermutationGroup(Integer(5)) >>> C = G.commutator(H) >>> C.is_isomorphic(AlternatingGroup(Integer(5))) True - An abelian group will have a trivial commutator. - sage: G = CyclicPermutationGroup(10) sage: G.commutator() Permutation Group with generators [()] - >>> from sage.all import * >>> G = CyclicPermutationGroup(Integer(10)) >>> G.commutator() Permutation Group with generators [()] - The quotient of a group by its commutator is always abelian. - sage: G = DihedralGroup(20) sage: C = G.commutator() sage: Q = G.quotient(C) sage: Q.is_abelian() True - >>> from sage.all import * >>> G = DihedralGroup(Integer(20)) >>> C = G.commutator() >>> Q = G.quotient(C) >>> Q.is_abelian() True - When forming commutators from two groups, the order of the groups does not matter. - sage: D = DihedralGroup(3) sage: S = SymmetricGroup(2) sage: C1 = D.commutator(S); C1 Permutation Group with generators [(1,2,3)] sage: C2 = S.commutator(D); C2 Permutation Group with generators [(1,3,2)] sage: C1 == C2 True - >>> from sage.all import * >>> D = DihedralGroup(Integer(3)) >>> S = SymmetricGroup(Integer(2)) >>> C1 = D.commutator(S); C1 Permutation Group with generators [(1,2,3)] >>> C2 = S.commutator(D); C2 Permutation Group with generators [(1,3,2)] >>> C1 == C2 True - This method calls two different functions in GAP, so this tests that their results are consistent. The commutator groups may have different generators, but the groups are equal. - sage: G = DiCyclicGroup(3) sage: C = G.commutator(); C Permutation Group with generators [(5,7,6)] sage: CC = G.commutator(G); CC Permutation Group with generators [(5,6,7)] sage: C == CC True - >>> from sage.all import * >>> G = DiCyclicGroup(Integer(3)) >>> C = G.commutator(); C Permutation Group with generators [(5,7,6)] >>> CC = G.commutator(G); CC Permutation Group with generators [(5,6,7)] >>> C == CC True - The second group is checked. - sage: G = SymmetricGroup(2) sage: G.commutator('junk') Traceback (most recent call last): ... TypeError: junk is not a permutation group - >>> from sage.all import * >>> G = SymmetricGroup(Integer(2)) >>> G.commutator('junk') Traceback (most recent call last): ... TypeError: junk is not a permutation group 
 - composition_series()[source]¶
- Return the composition series of this group as a list of permutation groups. - EXAMPLES: - These computations use pseudo-random numbers, so we set the seed for reproducible testing. - sage: set_random_seed(0) sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: G.composition_series() [Subgroup generated by [(3,4), (1,2,3)(4,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]), Subgroup generated by [(1,5)(3,4), (1,5)(2,4), (1,3,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]), Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])] sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]]) sage: CS = G.composition_series() sage: CS[3] Subgroup generated by [()] of (Permutation Group with generators [(1,2), (1,2,3)(4,5)]) - >>> from sage.all import * >>> set_random_seed(Integer(0)) >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]]) >>> G.composition_series() [Subgroup generated by [(3,4), (1,2,3)(4,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]), Subgroup generated by [(1,5)(3,4), (1,5)(2,4), (1,3,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]), Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])] >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))], [(Integer(1),Integer(2))]]) >>> CS = G.composition_series() >>> CS[Integer(3)] Subgroup generated by [()] of (Permutation Group with generators [(1,2), (1,2,3)(4,5)]) 
 - conjugacy_class(g)[source]¶
- Return the conjugacy class of - ginside the group- self.- INPUT: - g– an element of the permutation group- self
 - OUTPUT: - The conjugacy class of - gin the group- self. If- selfis the group denoted by \(G\), this method computes the set \(\{x^{-1}gx\ \vert\ x \in G \}\)- EXAMPLES: - sage: G = DihedralGroup(3) sage: g = G.gen(0) sage: G.conjugacy_class(g) Conjugacy class of (1,2,3) in Dihedral group of order 6 as a permutation group - >>> from sage.all import * >>> G = DihedralGroup(Integer(3)) >>> g = G.gen(Integer(0)) >>> G.conjugacy_class(g) Conjugacy class of (1,2,3) in Dihedral group of order 6 as a permutation group 
 - conjugacy_classes()[source]¶
- Return a list with all the conjugacy classes of - self.- EXAMPLES: - sage: G = DihedralGroup(3) sage: G.conjugacy_classes() [Conjugacy class of () in Dihedral group of order 6 as a permutation group, Conjugacy class of (2,3) in Dihedral group of order 6 as a permutation group, Conjugacy class of (1,2,3) in Dihedral group of order 6 as a permutation group] - >>> from sage.all import * >>> G = DihedralGroup(Integer(3)) >>> G.conjugacy_classes() [Conjugacy class of () in Dihedral group of order 6 as a permutation group, Conjugacy class of (2,3) in Dihedral group of order 6 as a permutation group, Conjugacy class of (1,2,3) in Dihedral group of order 6 as a permutation group] 
 - conjugacy_classes_representatives()[source]¶
- Return a complete list of representatives of conjugacy classes in a permutation group \(G\). - The ordering is that given by GAP. - EXAMPLES: - sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) sage: cl = G.conjugacy_classes_representatives(); cl [(), (2,4), (1,2)(3,4), (1,2,3,4), (1,3)(2,4)] sage: cl[3] in G True - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> cl = G.conjugacy_classes_representatives(); cl [(), (2,4), (1,2)(3,4), (1,2,3,4), (1,3)(2,4)] >>> cl[Integer(3)] in G True - sage: G = SymmetricGroup(5) sage: G.conjugacy_classes_representatives() # needs sage.combinat [(), (1,2), (1,2)(3,4), (1,2,3), (1,2,3)(4,5), (1,2,3,4), (1,2,3,4,5)] - >>> from sage.all import * >>> G = SymmetricGroup(Integer(5)) >>> G.conjugacy_classes_representatives() # needs sage.combinat [(), (1,2), (1,2)(3,4), (1,2,3), (1,2,3)(4,5), (1,2,3,4), (1,2,3,4,5)] - sage: S = SymmetricGroup(['a','b','c']) sage: S.conjugacy_classes_representatives() # needs sage.combinat [(), ('a','b'), ('a','b','c')] - >>> from sage.all import * >>> S = SymmetricGroup(['a','b','c']) >>> S.conjugacy_classes_representatives() # needs sage.combinat [(), ('a','b'), ('a','b','c')] - AUTHORS: - David Joyner and William Stein (2006-01-04) 
 
 - conjugacy_classes_subgroups()[source]¶
- Return a complete list of representatives of conjugacy classes of subgroups in a permutation group \(G\). - The ordering is that given by GAP. - EXAMPLES: - sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) sage: cl = G.conjugacy_classes_subgroups(); cl [Subgroup generated by [()] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(1,2)(3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(1,3)(2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(1,4)(2,3), (1,2)(3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(2,4), (1,3)(2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(1,2,3,4), (1,3)(2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(1,4)(2,3), (1,2)(3,4), (2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)])] - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> cl = G.conjugacy_classes_subgroups(); cl [Subgroup generated by [()] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(1,2)(3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(1,3)(2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(1,4)(2,3), (1,2)(3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(2,4), (1,3)(2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(1,2,3,4), (1,3)(2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]), Subgroup generated by [(1,4)(2,3), (1,2)(3,4), (2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)])] - sage: G = SymmetricGroup(3) sage: G.conjugacy_classes_subgroups() [Subgroup generated by [()] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(2,3)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,2,3)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,2,3), (2,3)] of (Symmetric group of order 3! as a permutation group)] - >>> from sage.all import * >>> G = SymmetricGroup(Integer(3)) >>> G.conjugacy_classes_subgroups() [Subgroup generated by [()] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(2,3)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,2,3)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,2,3), (2,3)] of (Symmetric group of order 3! as a permutation group)] - AUTHORS: - David Joyner (2006-10) 
 
 - conjugate(g)[source]¶
- Return the group formed by conjugating - selfwith- g.- INPUT: - g– a permutation group element, or an object that converts to a permutation group element, such as a list of integers or a string of cycles.
 - OUTPUT: - If - selfis the group denoted by \(H\), then this method computes the group\[g^{-1}Hg = \{g^{-1}hg\vert h\in H\}\]- which is the group \(H\) conjugated by \(g\). - There are no restrictions on - selfand- gbelonging to a common permutation group, and correspondingly, there is no relationship (such as a common parent) between- selfand the output group.- EXAMPLES: - sage: G = DihedralGroup(6) sage: a = PermutationGroupElement("(1,2,3,4)") sage: G.conjugate(a) Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] - >>> from sage.all import * >>> G = DihedralGroup(Integer(6)) >>> a = PermutationGroupElement("(1,2,3,4)") >>> G.conjugate(a) Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] - The element performing the conjugation can be specified in several ways. - sage: G = DihedralGroup(6) sage: strng = "(1,2,3,4)" sage: G.conjugate(strng) Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] sage: G = DihedralGroup(6) sage: lst = [2,3,4,1] sage: G.conjugate(lst) Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] sage: G = DihedralGroup(6) sage: cycles = [(1,2,3,4)] sage: G.conjugate(cycles) Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] - >>> from sage.all import * >>> G = DihedralGroup(Integer(6)) >>> strng = "(1,2,3,4)" >>> G.conjugate(strng) Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] >>> G = DihedralGroup(Integer(6)) >>> lst = [Integer(2),Integer(3),Integer(4),Integer(1)] >>> G.conjugate(lst) Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] >>> G = DihedralGroup(Integer(6)) >>> cycles = [(Integer(1),Integer(2),Integer(3),Integer(4))] >>> G.conjugate(cycles) Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)] - Conjugation is a group automorphism, so conjugate groups will be isomorphic. - sage: G = DiCyclicGroup(6) sage: G.degree() 11 sage: cycle = [i+1 for i in range(1,11)] + [1] sage: C = G.conjugate(cycle) sage: G.is_isomorphic(C) True - >>> from sage.all import * >>> G = DiCyclicGroup(Integer(6)) >>> G.degree() 11 >>> cycle = [i+Integer(1) for i in range(Integer(1),Integer(11))] + [Integer(1)] >>> C = G.conjugate(cycle) >>> G.is_isomorphic(C) True - The conjugating element may be from a symmetric group with larger degree than the group being conjugated. - sage: G = AlternatingGroup(5) sage: G.degree() 5 sage: g = "(1,3)(5,6,7)" sage: H = G.conjugate(g); H Permutation Group with generators [(1,4,6,3,2), (1,4,6)] sage: H.degree() 6 - >>> from sage.all import * >>> G = AlternatingGroup(Integer(5)) >>> G.degree() 5 >>> g = "(1,3)(5,6,7)" >>> H = G.conjugate(g); H Permutation Group with generators [(1,4,6,3,2), (1,4,6)] >>> H.degree() 6 - The conjugating element is checked. - sage: G = SymmetricGroup(3) sage: G.conjugate("junk") Traceback (most recent call last): ... TypeError: junk does not convert to a permutation group element - >>> from sage.all import * >>> G = SymmetricGroup(Integer(3)) >>> G.conjugate("junk") Traceback (most recent call last): ... TypeError: junk does not convert to a permutation group element 
 - construction()[source]¶
- Return the construction of - self.- EXAMPLES: - sage: P1 = PermutationGroup([[(1,2)]]) sage: P1.construction() (PermutationGroupFunctor[(1,2)], Permutation Group with generators [()]) sage: PermutationGroup([]).construction() is None True - >>> from sage.all import * >>> P1 = PermutationGroup([[(Integer(1),Integer(2))]]) >>> P1.construction() (PermutationGroupFunctor[(1,2)], Permutation Group with generators [()]) >>> PermutationGroup([]).construction() is None True - This allows us to perform computations like the following: - sage: P1 = PermutationGroup([[(1,2)]]); p1 = P1.gen() sage: P2 = PermutationGroup([[(1,3)]]); p2 = P2.gen() sage: p = p1*p2; p (1,2,3) sage: p.parent() Permutation Group with generators [(1,2), (1,3)] sage: p.parent().domain() {1, 2, 3} - >>> from sage.all import * >>> P1 = PermutationGroup([[(Integer(1),Integer(2))]]); p1 = P1.gen() >>> P2 = PermutationGroup([[(Integer(1),Integer(3))]]); p2 = P2.gen() >>> p = p1*p2; p (1,2,3) >>> p.parent() Permutation Group with generators [(1,2), (1,3)] >>> p.parent().domain() {1, 2, 3} - Note that this will merge permutation groups with different domains: - sage: g1 = PermutationGroupElement([(1,2),(3,4,5)]) sage: g2 = PermutationGroup([('a','b')], domain=['a', 'b']).gens()[0] sage: g2 ('a','b') sage: p = g1*g2; p (1,2)(3,4,5)('a','b') sage: P = parent(p) sage: P Permutation Group with generators [('a','b'), (1,2), (1,2,3,4,5)] - >>> from sage.all import * >>> g1 = PermutationGroupElement([(Integer(1),Integer(2)),(Integer(3),Integer(4),Integer(5))]) >>> g2 = PermutationGroup([('a','b')], domain=['a', 'b']).gens()[Integer(0)] >>> g2 ('a','b') >>> p = g1*g2; p (1,2)(3,4,5)('a','b') >>> P = parent(p) >>> P Permutation Group with generators [('a','b'), (1,2), (1,2,3,4,5)] 
 - cosets(S, side='right')[source]¶
- Return a list of the cosets of - Sin- self.- INPUT: - S– a subgroup of- self; an error is raised if- Sis not a subgroup
- side– (default:- 'right') determines if right cosets or left cosets are returned.- siderefers to where the representative is placed in the products forming the cosets and thus allowable values are only- 'right'and- 'left'.
 - OUTPUT: - A list of lists. Each inner list is a coset of the subgroup in the group. The first element of each coset is the smallest element (based on the ordering of the elements of - self) of all the group elements that have not yet appeared in a previous coset. The elements of each coset are in the same order as the subgroup elements used to build the coset’s elements.- As a consequence, the subgroup itself is the first coset, and its first element is the identity element. For each coset, the first element listed is the element used as a representative to build the coset. These representatives form an increasing sequence across the list of cosets, and within a coset the representative is the smallest element of its coset (both orderings are based on of the ordering of elements of - self).- In the case of a normal subgroup, left and right cosets should appear in the same order as part of the outer list. However, the list of the elements of a particular coset may be in a different order for the right coset versus the order in the left coset. So, if you check to see if a subgroup is normal, it is necessary to sort each individual coset first (but not the list of cosets, due to the ordering of the representatives). See below for examples of this. - Note - This is a naive implementation intended for instructional purposes, and hence is slow for larger groups. Sage and GAP provide more sophisticated functions for working quickly with cosets of larger groups. - EXAMPLES: - The default is to build right cosets. This example works with the symmetry group of an 8-gon and a normal subgroup. Notice that a straight check on the equality of the output is not sufficient to check normality, while sorting the individual cosets is sufficient to then simply test equality of the list of lists. Study the second coset in each list to understand the need for sorting the elements of the cosets. - sage: G = DihedralGroup(8) sage: quarter_turn = G('(1,3,5,7)(2,4,6,8)'); quarter_turn (1,3,5,7)(2,4,6,8) sage: S = G.subgroup([quarter_turn]) sage: rc = G.cosets(S); rc [[(), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,7,5,3)(2,8,6,4)], [(2,8)(3,7)(4,6), (1,7)(2,6)(3,5), (1,5)(2,4)(6,8), (1,3)(4,8)(5,7)], [(1,2)(3,8)(4,7)(5,6), (1,8)(2,7)(3,6)(4,5), (1,6)(2,5)(3,4)(7,8), (1,4)(2,3)(5,8)(6,7)], [(1,2,3,4,5,6,7,8), (1,4,7,2,5,8,3,6), (1,6,3,8,5,2,7,4), (1,8,7,6,5,4,3,2)]] sage: lc = G.cosets(S, side='left'); lc [[(), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,7,5,3)(2,8,6,4)], [(2,8)(3,7)(4,6), (1,3)(4,8)(5,7), (1,5)(2,4)(6,8), (1,7)(2,6)(3,5)], [(1,2)(3,8)(4,7)(5,6), (1,4)(2,3)(5,8)(6,7), (1,6)(2,5)(3,4)(7,8), (1,8)(2,7)(3,6)(4,5)], [(1,2,3,4,5,6,7,8), (1,4,7,2,5,8,3,6), (1,6,3,8,5,2,7,4), (1,8,7,6,5,4,3,2)]] sage: S.is_normal(G) True sage: rc == lc False sage: rc_sorted = [sorted(c) for c in rc] sage: lc_sorted = [sorted(c) for c in lc] sage: rc_sorted == lc_sorted True - >>> from sage.all import * >>> G = DihedralGroup(Integer(8)) >>> quarter_turn = G('(1,3,5,7)(2,4,6,8)'); quarter_turn (1,3,5,7)(2,4,6,8) >>> S = G.subgroup([quarter_turn]) >>> rc = G.cosets(S); rc [[(), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,7,5,3)(2,8,6,4)], [(2,8)(3,7)(4,6), (1,7)(2,6)(3,5), (1,5)(2,4)(6,8), (1,3)(4,8)(5,7)], [(1,2)(3,8)(4,7)(5,6), (1,8)(2,7)(3,6)(4,5), (1,6)(2,5)(3,4)(7,8), (1,4)(2,3)(5,8)(6,7)], [(1,2,3,4,5,6,7,8), (1,4,7,2,5,8,3,6), (1,6,3,8,5,2,7,4), (1,8,7,6,5,4,3,2)]] >>> lc = G.cosets(S, side='left'); lc [[(), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,7,5,3)(2,8,6,4)], [(2,8)(3,7)(4,6), (1,3)(4,8)(5,7), (1,5)(2,4)(6,8), (1,7)(2,6)(3,5)], [(1,2)(3,8)(4,7)(5,6), (1,4)(2,3)(5,8)(6,7), (1,6)(2,5)(3,4)(7,8), (1,8)(2,7)(3,6)(4,5)], [(1,2,3,4,5,6,7,8), (1,4,7,2,5,8,3,6), (1,6,3,8,5,2,7,4), (1,8,7,6,5,4,3,2)]] >>> S.is_normal(G) True >>> rc == lc False >>> rc_sorted = [sorted(c) for c in rc] >>> lc_sorted = [sorted(c) for c in lc] >>> rc_sorted == lc_sorted True - An example with the symmetry group of a regular tetrahedron and a subgroup that is not normal. Thus, the right and left cosets are different (and so are the representatives). With each individual coset sorted, a naive test of normality is possible. - sage: A = AlternatingGroup(4) sage: face_turn = A('(1,2,3)'); face_turn (1,2,3) sage: stabilizer = A.subgroup([face_turn]) sage: rc = A.cosets(stabilizer, side='right'); rc [[(), (1,2,3), (1,3,2)], [(2,3,4), (1,3)(2,4), (1,4,2)], [(2,4,3), (1,4,3), (1,2)(3,4)], [(1,2,4), (1,4)(2,3), (1,3,4)]] sage: lc = A.cosets(stabilizer, side='left'); lc [[(), (1,2,3), (1,3,2)], [(2,3,4), (1,2)(3,4), (1,3,4)], [(2,4,3), (1,2,4), (1,3)(2,4)], [(1,4,2), (1,4,3), (1,4)(2,3)]] sage: stabilizer.is_normal(A) False sage: rc_sorted = [sorted(c) for c in rc] sage: lc_sorted = [sorted(c) for c in lc] sage: rc_sorted == lc_sorted False - >>> from sage.all import * >>> A = AlternatingGroup(Integer(4)) >>> face_turn = A('(1,2,3)'); face_turn (1,2,3) >>> stabilizer = A.subgroup([face_turn]) >>> rc = A.cosets(stabilizer, side='right'); rc [[(), (1,2,3), (1,3,2)], [(2,3,4), (1,3)(2,4), (1,4,2)], [(2,4,3), (1,4,3), (1,2)(3,4)], [(1,2,4), (1,4)(2,3), (1,3,4)]] >>> lc = A.cosets(stabilizer, side='left'); lc [[(), (1,2,3), (1,3,2)], [(2,3,4), (1,2)(3,4), (1,3,4)], [(2,4,3), (1,2,4), (1,3)(2,4)], [(1,4,2), (1,4,3), (1,4)(2,3)]] >>> stabilizer.is_normal(A) False >>> rc_sorted = [sorted(c) for c in rc] >>> lc_sorted = [sorted(c) for c in lc] >>> rc_sorted == lc_sorted False - AUTHOR: - Rob Beezer (2011-01-31) 
 
 - degree()[source]¶
- Return the degree of this permutation group. - EXAMPLES: - sage: S = SymmetricGroup(['a','b','c']) sage: S.degree() 3 sage: G = PermutationGroup([(1,3),(4,5)]) sage: G.degree() 5 - >>> from sage.all import * >>> S = SymmetricGroup(['a','b','c']) >>> S.degree() 3 >>> G = PermutationGroup([(Integer(1),Integer(3)),(Integer(4),Integer(5))]) >>> G.degree() 5 - Note that you can explicitly specify the domain to get a permutation group of smaller degree: - sage: G = PermutationGroup([(1,3),(4,5)], domain=[1,3,4,5]) sage: G.degree() 4 - >>> from sage.all import * >>> G = PermutationGroup([(Integer(1),Integer(3)),(Integer(4),Integer(5))], domain=[Integer(1),Integer(3),Integer(4),Integer(5)]) >>> G.degree() 4 
 - derived_series()[source]¶
- Return the derived series of this group as a list of permutation groups. - EXAMPLES: - These computations use pseudo-random numbers, so we set the seed for reproducible testing. - sage: set_random_seed(0) sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: G.derived_series() [Subgroup generated by [(3,4), (1,2,3)(4,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]), Subgroup generated by [(1,5)(3,4), (1,5)(2,4), (1,3,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])] - >>> from sage.all import * >>> set_random_seed(Integer(0)) >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]]) >>> G.derived_series() [Subgroup generated by [(3,4), (1,2,3)(4,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]), Subgroup generated by [(1,5)(3,4), (1,5)(2,4), (1,3,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])] 
 - direct_product(other, maps=True)[source]¶
- Wraps GAP’s - DirectProduct,- Embedding, and- Projection.- Sage calls GAP’s - DirectProduct, which chooses an efficient representation for the direct product. The direct product of permutation groups will be a permutation group again. For a direct product- D, the GAP operation- Embedding(D,i)returns the homomorphism embedding the \(i\)-th factor into- D. The GAP operation- Projection(D,i)gives the projection of- Donto the \(i\)-th factor. This method returns a 5-tuple: a permutation group and 4 morphisms.- INPUT: - self,- other– permutation groups
 - OUTPUT: - D– a direct product of the inputs, returned as a permutation group as well
- iota1– an embedding of- selfinto- D
- iota2– an embedding of- otherinto- D
- pr1– the projection of- Donto- self(giving a splitting- 1 - other - D - self - 1)
- pr2– the projection of- Donto- other(giving a splitting- 1 - self - D - other - 1)
 - EXAMPLES: - sage: G = CyclicPermutationGroup(4) sage: D = G.direct_product(G, False); D Permutation Group with generators [(1,2,3,4), (5,6,7,8)] sage: D,iota1,iota2,pr1,pr2 = G.direct_product(G) sage: D; iota1; iota2; pr1; pr2 Permutation Group with generators [(1,2,3,4), (5,6,7,8)] Permutation group morphism: From: Cyclic group of order 4 as a permutation group To: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] Defn: Embedding( Group( [ (1,2,3,4), (5,6,7,8) ] ), 1 ) Permutation group morphism: From: Cyclic group of order 4 as a permutation group To: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] Defn: Embedding( Group( [ (1,2,3,4), (5,6,7,8) ] ), 2 ) Permutation group morphism: From: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] To: Cyclic group of order 4 as a permutation group Defn: Projection( Group( [ (1,2,3,4), (5,6,7,8) ] ), 1 ) Permutation group morphism: From: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] To: Cyclic group of order 4 as a permutation group Defn: Projection( Group( [ (1,2,3,4), (5,6,7,8) ] ), 2 ) sage: g = D([(1,3),(2,4)]); g (1,3)(2,4) sage: d = D([(1,4,3,2),(5,7),(6,8)]); d (1,4,3,2)(5,7)(6,8) sage: iota1(g); iota2(g); pr1(d); pr2(d) (1,3)(2,4) (5,7)(6,8) (1,4,3,2) (1,3)(2,4) - >>> from sage.all import * >>> G = CyclicPermutationGroup(Integer(4)) >>> D = G.direct_product(G, False); D Permutation Group with generators [(1,2,3,4), (5,6,7,8)] >>> D,iota1,iota2,pr1,pr2 = G.direct_product(G) >>> D; iota1; iota2; pr1; pr2 Permutation Group with generators [(1,2,3,4), (5,6,7,8)] Permutation group morphism: From: Cyclic group of order 4 as a permutation group To: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] Defn: Embedding( Group( [ (1,2,3,4), (5,6,7,8) ] ), 1 ) Permutation group morphism: From: Cyclic group of order 4 as a permutation group To: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] Defn: Embedding( Group( [ (1,2,3,4), (5,6,7,8) ] ), 2 ) Permutation group morphism: From: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] To: Cyclic group of order 4 as a permutation group Defn: Projection( Group( [ (1,2,3,4), (5,6,7,8) ] ), 1 ) Permutation group morphism: From: Permutation Group with generators [(1,2,3,4), (5,6,7,8)] To: Cyclic group of order 4 as a permutation group Defn: Projection( Group( [ (1,2,3,4), (5,6,7,8) ] ), 2 ) >>> g = D([(Integer(1),Integer(3)),(Integer(2),Integer(4))]); g (1,3)(2,4) >>> d = D([(Integer(1),Integer(4),Integer(3),Integer(2)),(Integer(5),Integer(7)),(Integer(6),Integer(8))]); d (1,4,3,2)(5,7)(6,8) >>> iota1(g); iota2(g); pr1(d); pr2(d) (1,3)(2,4) (5,7)(6,8) (1,4,3,2) (1,3)(2,4) 
 - disjoint_direct_product_decomposition()[source]¶
- Return the finest partition of the underlying set such that - selfis isomorphic to the direct product of the projections of- selfonto each part of the partition. Each part is a union of orbits of- self.- The algorithm is from [CJ2022], which runs in time polynomial in \(n \cdot |X|\), where \(n\) is the degree of the group and \(|X|\) is the size of a generating set, see Theorem 4.5. - EXAMPLES: - The example from the original paper: - sage: H = PermutationGroup([[(1,2,3),(7,9,8),(10,12,11)],[(4,5,6),(7,8,9),(10,11,12)],[(5,6),(8,9),(11,12)],[(7,8,9),(10,11,12)]]) sage: S = H.disjoint_direct_product_decomposition(); S {{1, 2, 3}, {4, 5, 6, 7, 8, 9, 10, 11, 12}} sage: A = libgap.Stabilizer(H, list(S[0]), libgap.OnTuples); A Group([ (7,8,9)(10,11,12), (5,6)(8,9)(11,12), (4,5,6)(7,8,9)(10,11,12) ]) sage: B = libgap.Stabilizer(H, list(S[1]), libgap.OnTuples); B Group([ (1,2,3) ]) sage: T = PermutationGroup(gap_group=libgap.DirectProduct(A,B)) sage: T.is_isomorphic(H) True sage: PermutationGroup(PermutationGroup(gap_group=A).gens(),domain=list(S[1])).disjoint_direct_product_decomposition() {{4, 5, 6, 7, 8, 9, 10, 11, 12}} sage: PermutationGroup(PermutationGroup(gap_group=B).gens(),domain=list(S[0])).disjoint_direct_product_decomposition() {{1, 2, 3}} - >>> from sage.all import * >>> H = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(7),Integer(9),Integer(8)),(Integer(10),Integer(12),Integer(11))],[(Integer(4),Integer(5),Integer(6)),(Integer(7),Integer(8),Integer(9)),(Integer(10),Integer(11),Integer(12))],[(Integer(5),Integer(6)),(Integer(8),Integer(9)),(Integer(11),Integer(12))],[(Integer(7),Integer(8),Integer(9)),(Integer(10),Integer(11),Integer(12))]]) >>> S = H.disjoint_direct_product_decomposition(); S {{1, 2, 3}, {4, 5, 6, 7, 8, 9, 10, 11, 12}} >>> A = libgap.Stabilizer(H, list(S[Integer(0)]), libgap.OnTuples); A Group([ (7,8,9)(10,11,12), (5,6)(8,9)(11,12), (4,5,6)(7,8,9)(10,11,12) ]) >>> B = libgap.Stabilizer(H, list(S[Integer(1)]), libgap.OnTuples); B Group([ (1,2,3) ]) >>> T = PermutationGroup(gap_group=libgap.DirectProduct(A,B)) >>> T.is_isomorphic(H) True >>> PermutationGroup(PermutationGroup(gap_group=A).gens(),domain=list(S[Integer(1)])).disjoint_direct_product_decomposition() {{4, 5, 6, 7, 8, 9, 10, 11, 12}} >>> PermutationGroup(PermutationGroup(gap_group=B).gens(),domain=list(S[Integer(0)])).disjoint_direct_product_decomposition() {{1, 2, 3}} - An example with a different domain: - sage: PermutationGroup([[('a','c','d'),('b','e')]]).disjoint_direct_product_decomposition() {{'a', 'c', 'd'}, {'b', 'e'}} sage: PermutationGroup([[('a','c','d','b','e')]]).disjoint_direct_product_decomposition() {{'a', 'b', 'c', 'd', 'e'}} - >>> from sage.all import * >>> PermutationGroup([[('a','c','d'),('b','e')]]).disjoint_direct_product_decomposition() {{'a', 'c', 'd'}, {'b', 'e'}} >>> PermutationGroup([[('a','c','d','b','e')]]).disjoint_direct_product_decomposition() {{'a', 'b', 'c', 'd', 'e'}} - Counting the number of “connected” permutation groups of degree \(n\): - sage: seq = [sum(1 for G in SymmetricGroup(n).conjugacy_classes_subgroups() if len(G.disjoint_direct_product_decomposition()) == 1) for n in range(1,8)]; seq [1, 1, 2, 6, 6, 27, 20] sage: oeis(seq) # optional -- internet 0: A005226: Number of atomic species of degree n; also number of connected permutation groups of degree n. - >>> from sage.all import * >>> seq = [sum(Integer(1) for G in SymmetricGroup(n).conjugacy_classes_subgroups() if len(G.disjoint_direct_product_decomposition()) == Integer(1)) for n in range(Integer(1),Integer(8))]; seq [1, 1, 2, 6, 6, 27, 20] >>> oeis(seq) # optional -- internet 0: A005226: Number of atomic species of degree n; also number of connected permutation groups of degree n. 
 - domain()[source]¶
- Return the underlying set that this permutation group acts on. - EXAMPLES: - sage: P = PermutationGroup([(1,2),(3,5)]) sage: P.domain() {1, 2, 3, 4, 5} sage: S = SymmetricGroup(['a', 'b', 'c']) sage: S.domain() {'a', 'b', 'c'} - >>> from sage.all import * >>> P = PermutationGroup([(Integer(1),Integer(2)),(Integer(3),Integer(5))]) >>> P.domain() {1, 2, 3, 4, 5} >>> S = SymmetricGroup(['a', 'b', 'c']) >>> S.domain() {'a', 'b', 'c'} 
 - exponent()[source]¶
- Compute the exponent of the group. - The exponent \(e\) of a group \(G\) is the LCM of the orders of its elements, that is, \(e\) is the smallest integer such that \(g^e=1\) for all \(g \in G\). - EXAMPLES: - sage: G = AlternatingGroup(4) sage: G.exponent() 6 - >>> from sage.all import * >>> G = AlternatingGroup(Integer(4)) >>> G.exponent() 6 
 - fitting_subgroup()[source]¶
- Return the Fitting subgroup of - self.- The Fitting subgroup of a group \(G\) is the largest nilpotent normal subgroup of \(G\). - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]]) sage: G.fitting_subgroup() Subgroup generated by [(2,4), (1,2,3,4), (1,3)] of (Permutation Group with generators [(2,4), (1,2,3,4)]) sage: G = PermutationGroup([[(1,2,3,4)],[(1,2)]]) sage: G.fitting_subgroup() Subgroup generated by [(1,2)(3,4), (1,3)(2,4)] of (Permutation Group with generators [(1,2), (1,2,3,4)]) - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(2),Integer(4))]]) >>> G.fitting_subgroup() Subgroup generated by [(2,4), (1,2,3,4), (1,3)] of (Permutation Group with generators [(2,4), (1,2,3,4)]) >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(1),Integer(2))]]) >>> G.fitting_subgroup() Subgroup generated by [(1,2)(3,4), (1,3)(2,4)] of (Permutation Group with generators [(1,2), (1,2,3,4)]) 
 - fixed_points()[source]¶
- Return the list of points fixed by - self, i.e., the subset of- .domain()not moved by any element of- self.- EXAMPLES: - sage: G = PermutationGroup([(1,2,3)]) sage: G.fixed_points() [] sage: G = PermutationGroup([(1,2,3),(5,6)]) sage: G.fixed_points() [4] sage: G = PermutationGroup([[(1,4,7)],[(4,3),(6,7)]]) sage: G.fixed_points() [2, 5] - >>> from sage.all import * >>> G = PermutationGroup([(Integer(1),Integer(2),Integer(3))]) >>> G.fixed_points() [] >>> G = PermutationGroup([(Integer(1),Integer(2),Integer(3)),(Integer(5),Integer(6))]) >>> G.fixed_points() [4] >>> G = PermutationGroup([[(Integer(1),Integer(4),Integer(7))],[(Integer(4),Integer(3)),(Integer(6),Integer(7))]]) >>> G.fixed_points() [2, 5] 
 - frattini_subgroup()[source]¶
- Return the Frattini subgroup of - self.- The Frattini subgroup of a group \(G\) is the intersection of all maximal subgroups of \(G\). - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]]) sage: G.frattini_subgroup() Subgroup generated by [(1,3)(2,4)] of (Permutation Group with generators [(2,4), (1,2,3,4)]) sage: G = SymmetricGroup(4) sage: G.frattini_subgroup() Subgroup generated by [()] of (Symmetric group of order 4! as a permutation group) - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(2),Integer(4))]]) >>> G.frattini_subgroup() Subgroup generated by [(1,3)(2,4)] of (Permutation Group with generators [(2,4), (1,2,3,4)]) >>> G = SymmetricGroup(Integer(4)) >>> G.frattini_subgroup() Subgroup generated by [()] of (Symmetric group of order 4! as a permutation group) 
 - gap()[source]¶
- This method from - sage.groups.libgap_wrapper.ParentLibGAPis added in order to achieve compatibility and have- sage.groups.libgap_morphism.GroupHomset_libgapwork for permutation groups, as well- OUTPUT: an instance of - sage.libs.gap.element.GapElementrepresenting this group- EXAMPLES: - sage: P8 = PSp(8,3) sage: P8.gap() <permutation group of size 65784756654489600 with 2 generators> sage: gap(P8) == P8.gap() False sage: S3 = SymmetricGroup(3) sage: S3.gap() Sym( [ 1 .. 3 ] ) sage: gap(S3) == S3.gap() False - >>> from sage.all import * >>> P8 = PSp(Integer(8),Integer(3)) >>> P8.gap() <permutation group of size 65784756654489600 with 2 generators> >>> gap(P8) == P8.gap() False >>> S3 = SymmetricGroup(Integer(3)) >>> S3.gap() Sym( [ 1 .. 3 ] ) >>> gap(S3) == S3.gap() False 
 - gen(i=None)[source]¶
- Return the \(i\)-th generator of - self; that is, the \(i\)-th element of the list- self.gens().- The argument \(i\) may be omitted if there is only one generator (but this will raise an error otherwise). - EXAMPLES: - We explicitly construct the alternating group on four elements: - sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4 Permutation Group with generators [(2,3,4), (1,2,3)] sage: A4.gens() ((2,3,4), (1,2,3)) sage: A4.gen(0) (2,3,4) sage: A4.gen(1) (1,2,3) sage: A4.gens()[0]; A4.gens()[1] (2,3,4) (1,2,3) sage: P1 = PermutationGroup([[(1,2)]]); P1.gen() (1,2) - >>> from sage.all import * >>> A4 = PermutationGroup([[(Integer(1),Integer(2),Integer(3))],[(Integer(2),Integer(3),Integer(4))]]); A4 Permutation Group with generators [(2,3,4), (1,2,3)] >>> A4.gens() ((2,3,4), (1,2,3)) >>> A4.gen(Integer(0)) (2,3,4) >>> A4.gen(Integer(1)) (1,2,3) >>> A4.gens()[Integer(0)]; A4.gens()[Integer(1)] (2,3,4) (1,2,3) >>> P1 = PermutationGroup([[(Integer(1),Integer(2))]]); P1.gen() (1,2) 
 - gens()[source]¶
- Return tuple of generators of this group. - These need not be minimal, as they are the generators used in defining this group. - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3)], [(1,2)]]) sage: G.gens() ((1,2), (1,2,3)) - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3))], [(Integer(1),Integer(2))]]) >>> G.gens() ((1,2), (1,2,3)) - Note that the generators need not be minimal, though duplicates are removed: - sage: G = PermutationGroup([[(1,2)], [(1,3)], [(2,3)], [(1,2)]]) sage: G.gens() ((2,3), (1,2), (1,3)) - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2))], [(Integer(1),Integer(3))], [(Integer(2),Integer(3))], [(Integer(1),Integer(2))]]) >>> G.gens() ((2,3), (1,2), (1,3)) - We can use index notation to access the generators returned by - self.gens:- sage: G = PermutationGroup([[(1,2,3,4), (5,6)], [(1,2)]]) sage: g = G.gens() sage: g[0] (1,2) sage: g[1] (1,2,3,4)(5,6) - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4)), (Integer(5),Integer(6))], [(Integer(1),Integer(2))]]) >>> g = G.gens() >>> g[Integer(0)] (1,2) >>> g[Integer(1)] (1,2,3,4)(5,6) 
 - gens_small()[source]¶
- For this group, returns a generating set which has few elements. - As neither irredundancy nor minimal length is proven, it is fast. - EXAMPLES: - sage: R = "(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24)" ## R = right sage: U = "( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19)" ## U = top sage: L = "( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35)" ## L = left sage: F = "(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11)" ## F = front sage: B = "(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27)" ## B = back or rear sage: D = "(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40)" ## D = down or bottom sage: G = PermutationGroup([R,L,U,F,B,D]) sage: len(G.gens_small()) 2 - >>> from sage.all import * >>> R = "(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24)" ## R = right >>> U = "( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19)" ## U = top >>> L = "( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35)" ## L = left >>> F = "(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11)" ## F = front >>> B = "(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27)" ## B = back or rear >>> D = "(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40)" ## D = down or bottom >>> G = PermutationGroup([R,L,U,F,B,D]) >>> len(G.gens_small()) 2 - The output may be unpredictable, due to the use of randomized algorithms in GAP. Note that both the following answers are equally valid. - sage: G = PermutationGroup([[('a','b')], [('b', 'c')], [('a', 'c')]]) sage: G.gens_small() # random [('b','c'), ('a','c','b')] ## (on 64-bit Linux) [('a','b'), ('a','c','b')] ## (on Solaris) sage: len(G.gens_small()) == 2 # random True - >>> from sage.all import * >>> G = PermutationGroup([[('a','b')], [('b', 'c')], [('a', 'c')]]) >>> G.gens_small() # random [('b','c'), ('a','c','b')] ## (on 64-bit Linux) [('a','b'), ('a','c','b')] ## (on Solaris) >>> len(G.gens_small()) == Integer(2) # random True 
 - group_id()[source]¶
- Return the ID code of this group, which is a list of two integers. - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]]) sage: G.group_id() [12, 4] - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))], [(Integer(1),Integer(2))]]) >>> G.group_id() [12, 4] 
 - group_primitive_id()[source]¶
- Return the index of this group in the GAP database of primitive groups. - OUTPUT: - A positive integer, following GAP’s conventions. A - ValueErroris raised if the group is not primitive.- EXAMPLES: - sage: G = PermutationGroup([[(1,2,3,4,5)], [(1,5),(2,4)]]) sage: G.group_primitive_id() 2 sage: G.degree() 5 - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4),Integer(5))], [(Integer(1),Integer(5)),(Integer(2),Integer(4))]]) >>> G.group_primitive_id() 2 >>> G.degree() 5 - From the information of the degree and the identification number, you can recover the isomorphism class of your group in the GAP database: - sage: H = PrimitiveGroup(5,2) sage: G == H False sage: G.is_isomorphic(H) True - >>> from sage.all import * >>> H = PrimitiveGroup(Integer(5),Integer(2)) >>> G == H False >>> G.is_isomorphic(H) True 
 - has_regular_subgroup(return_group=False)[source]¶
- Return whether the group contains a regular subgroup. - INPUT: - return_group– boolean; if- True, a regular subgroup is returned if there is one, and- Noneif there isn’t. When- return_group=False(default), only a boolean indicating whether such a group exists is returned instead.
 - EXAMPLES: - The symmetric group on 4 elements has a regular subgroup: - sage: S4 = groups.permutation.Symmetric(4) sage: S4.has_regular_subgroup() True sage: S4.has_regular_subgroup(return_group=True) # random Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,3)(2,4), (1,4)(2,3)] - >>> from sage.all import * >>> S4 = groups.permutation.Symmetric(Integer(4)) >>> S4.has_regular_subgroup() True >>> S4.has_regular_subgroup(return_group=True) # random Subgroup of (Symmetric group of order 4! as a permutation group) generated by [(1,3)(2,4), (1,4)(2,3)] - But the automorphism group of Petersen’s graph does not: - sage: # needs sage.graphs sage: G = graphs.PetersenGraph().automorphism_group() sage: G.has_regular_subgroup() False - >>> from sage.all import * >>> # needs sage.graphs >>> G = graphs.PetersenGraph().automorphism_group() >>> G.has_regular_subgroup() False 
 - holomorph()[source]¶
- The holomorph of a group as a permutation group. - The holomorph of a group \(G\) is the semidirect product \(G \rtimes_{id} Aut(G)\), where \(id\) is the identity function on \(Aut(G)\), the automorphism group of \(G\). - See Wikipedia article Holomorph (mathematics) - OUTPUT: the holomorph of a given group as permutation group via a wrapping of GAP’s semidirect product function - EXAMPLES: - Thomas and Wood’s ‘Group Tables’ (Shiva Publishing, 1980) tells us that the holomorph of \(C_5\) is the unique group of order 20 with a trivial center. - sage: C5 = CyclicPermutationGroup(5) sage: A = C5.holomorph() sage: A.order() 20 sage: A.is_abelian() False sage: A.center() Subgroup generated by [()] of (Permutation Group with generators [(5,6,7,8,9), (1,2,4,3)(6,7,9,8)]) sage: A Permutation Group with generators [(5,6,7,8,9), (1,2,4,3)(6,7,9,8)] - >>> from sage.all import * >>> C5 = CyclicPermutationGroup(Integer(5)) >>> A = C5.holomorph() >>> A.order() 20 >>> A.is_abelian() False >>> A.center() Subgroup generated by [()] of (Permutation Group with generators [(5,6,7,8,9), (1,2,4,3)(6,7,9,8)]) >>> A Permutation Group with generators [(5,6,7,8,9), (1,2,4,3)(6,7,9,8)] - Noting that the automorphism group of \(D_4\) is itself \(D_4\), it can easily be shown that the holomorph is indeed an internal semidirect product of these two groups. - sage: D4 = DihedralGroup(4) sage: H = D4.holomorph() sage: H.gens() ((2,3,5,8), (2,5)(3,8), (3,8)(4,7), (1,4,6,7)(2,3,5,8), (1,8)(2,7)(3,6)(4,5)) sage: G = H.subgroup([H.gens()[0],H.gens()[1],H.gens()[2]]) sage: N = H.subgroup([H.gens()[3],H.gens()[4]]) sage: N.is_normal(H) True sage: G.is_isomorphic(D4) True sage: N.is_isomorphic(D4) True sage: G.intersection(N) Permutation Group with generators [()] sage: L = [H(x)*H(y) for x in G for y in N]; L.sort() sage: L1 = H.list(); L1.sort() sage: L == L1 True - >>> from sage.all import * >>> D4 = DihedralGroup(Integer(4)) >>> H = D4.holomorph() >>> H.gens() ((2,3,5,8), (2,5)(3,8), (3,8)(4,7), (1,4,6,7)(2,3,5,8), (1,8)(2,7)(3,6)(4,5)) >>> G = H.subgroup([H.gens()[Integer(0)],H.gens()[Integer(1)],H.gens()[Integer(2)]]) >>> N = H.subgroup([H.gens()[Integer(3)],H.gens()[Integer(4)]]) >>> N.is_normal(H) True >>> G.is_isomorphic(D4) True >>> N.is_isomorphic(D4) True >>> G.intersection(N) Permutation Group with generators [()] >>> L = [H(x)*H(y) for x in G for y in N]; L.sort() >>> L1 = H.list(); L1.sort() >>> L == L1 True - Author: - Kevin Halasz (2012-08-14) 
 
 - homology(n, p=0)[source]¶
- Compute the group homology \(H_n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. Wraps HAP’s - GroupHomologyfunction, written by Graham Ellis.- REQUIRES: GAP package HAP (in gap_packages-*.spkg). - AUTHORS: - David Joyner and Graham Ellis 
 - The example below computes \(H_7(S_5, \ZZ)\), \(H_7(S_5, \ZZ / 2 \ZZ)\), \(H_7(S_5, \ZZ / 3 \ZZ)\), and \(H_7(S_5, \ZZ / 5 \ZZ)\), respectively. To compute the \(2\)-part of \(H_7(S_5, \ZZ)\), use the method - homology_part().- EXAMPLES: - sage: G = SymmetricGroup(5) sage: G.homology(7) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 x C2 x C4 x C3 x C5 sage: G.homology(7,2) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 x C2 x C2 x C2 x C2 sage: G.homology(7,3) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C3 sage: G.homology(7,5) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C5 - >>> from sage.all import * >>> G = SymmetricGroup(Integer(5)) >>> G.homology(Integer(7)) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 x C2 x C4 x C3 x C5 >>> G.homology(Integer(7),Integer(2)) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 x C2 x C2 x C2 x C2 >>> G.homology(Integer(7),Integer(3)) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C3 >>> G.homology(Integer(7),Integer(5)) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C5 - REFERENCES: - G. Ellis, “Computing group resolutions”, J. Symbolic Computation. Vol.38, (2004)1077-1118 (Available at http://hamilton.nuigalway.ie/. 
- D. Joyner, “A primer on computational group homology and cohomology”, http://front.math.ucdavis.edu/0706.0549 
 
 - homology_part(n, p=0)[source]¶
- Compute the \(p\)-part of the group homology \(H_n(G, F)\), where \(F = \ZZ\) if \(p=0\) and \(F = \ZZ / p \ZZ\) if \(p > 0\) is a prime. Wraps HAP’s - Homologyfunction, written by Graham Ellis, applied to the \(p\)-Sylow subgroup of \(G\).- REQUIRES: GAP package HAP (in gap_packages-*.spkg). - EXAMPLES: - sage: G = SymmetricGroup(5) sage: G.homology_part(7,2) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 x C2 x C2 x C2 x C4 - >>> from sage.all import * >>> G = SymmetricGroup(Integer(5)) >>> G.homology_part(Integer(7),Integer(2)) # optional - gap_package_hap Multiplicative Abelian group isomorphic to C2 x C2 x C2 x C2 x C4 - AUTHORS: - David Joyner and Graham Ellis 
 
 - id()[source]¶
- Return the ID code of this group, which is a list of two integers. - Same as - group_id().- EXAMPLES: - sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]]) sage: G.group_id() [12, 4] - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))], [(Integer(1),Integer(2))]]) >>> G.group_id() [12, 4] 
 - identity()[source]¶
- Return the identity element of this group. - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3),(4,5)]]) sage: e = G.identity(); e # indirect doctest () sage: g = G.gen(0) sage: g*e (1,2,3)(4,5) sage: e*g (1,2,3)(4,5) sage: S = SymmetricGroup(['a','b','c']) sage: S.identity() () - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))]]) >>> e = G.identity(); e # indirect doctest () >>> g = G.gen(Integer(0)) >>> g*e (1,2,3)(4,5) >>> e*g (1,2,3)(4,5) >>> S = SymmetricGroup(['a','b','c']) >>> S.identity() () 
 - intersection(other)[source]¶
- Return the permutation group that is the intersection of - selfand- other.- INPUT: - other– a permutation group
 - OUTPUT: - A permutation group that is the set-theoretic intersection of - selfwith- other. The groups are viewed as subgroups of a symmetric group big enough to contain both group’s symbol sets. So there is no strict notion of the two groups being subgroups of a common parent.- EXAMPLES: - sage: H = DihedralGroup(4) sage: K = CyclicPermutationGroup(4) sage: H.intersection(K) Permutation Group with generators [(1,2,3,4)] sage: L = DihedralGroup(5) sage: H.intersection(L) Permutation Group with generators [(1,4)(2,3)] sage: M = PermutationGroup(["()"]) sage: H.intersection(M) Permutation Group with generators [()] - >>> from sage.all import * >>> H = DihedralGroup(Integer(4)) >>> K = CyclicPermutationGroup(Integer(4)) >>> H.intersection(K) Permutation Group with generators [(1,2,3,4)] >>> L = DihedralGroup(Integer(5)) >>> H.intersection(L) Permutation Group with generators [(1,4)(2,3)] >>> M = PermutationGroup(["()"]) >>> H.intersection(M) Permutation Group with generators [()] - Some basic properties. - sage: H = DihedralGroup(4) sage: L = DihedralGroup(5) sage: H.intersection(L) == L.intersection(H) True sage: H.intersection(H) == H True - >>> from sage.all import * >>> H = DihedralGroup(Integer(4)) >>> L = DihedralGroup(Integer(5)) >>> H.intersection(L) == L.intersection(H) True >>> H.intersection(H) == H True - The group - otheris verified as such.- sage: H = DihedralGroup(4) sage: H.intersection('junk') Traceback (most recent call last): ... TypeError: junk is not a permutation group - >>> from sage.all import * >>> H = DihedralGroup(Integer(4)) >>> H.intersection('junk') Traceback (most recent call last): ... TypeError: junk is not a permutation group 
 - irreducible_characters()[source]¶
- Return a list of the irreducible characters of - self.- EXAMPLES: - sage: irr = SymmetricGroup(3).irreducible_characters() # needs sage.rings.number_field sage: [x.values() for x in irr] # needs sage.rings.number_field [[1, -1, 1], [2, 0, -1], [1, 1, 1]] - >>> from sage.all import * >>> irr = SymmetricGroup(Integer(3)).irreducible_characters() # needs sage.rings.number_field >>> [x.values() for x in irr] # needs sage.rings.number_field [[1, -1, 1], [2, 0, -1], [1, 1, 1]] 
 - is_abelian()[source]¶
- Return - Trueif this group is abelian.- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) sage: G.is_abelian() False sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_abelian() True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) >>> G.is_abelian() False >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_abelian() True 
 - is_commutative()[source]¶
- Return - Trueif this group is commutative.- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) sage: G.is_commutative() False sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_commutative() True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) >>> G.is_commutative() False >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_commutative() True 
 - is_cyclic()[source]¶
- Return - Trueif this group is cyclic.- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) sage: G.is_cyclic() False sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_cyclic() True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) >>> G.is_cyclic() False >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_cyclic() True 
 - is_elementary_abelian()[source]¶
- Return - Trueif this group is elementary abelian. An elementary abelian group is a finite abelian group, where every nontrivial element has order \(p\), where \(p\) is a prime.- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) sage: G.is_elementary_abelian() False sage: G = PermutationGroup(['(1,2,3)','(4,5,6)']) sage: G.is_elementary_abelian() True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) >>> G.is_elementary_abelian() False >>> G = PermutationGroup(['(1,2,3)','(4,5,6)']) >>> G.is_elementary_abelian() True 
 - is_isomorphic(right)[source]¶
- Return - Trueif the groups are isomorphic.- INPUT: - self– this group
- right– a permutation group
 - OUTPUT: boolean; - Trueif- selfand- rightare isomorphic groups;- Falseotherwise- EXAMPLES: - sage: v = ['(1,2,3)(4,5)', '(1,2,3,4,5)'] sage: G = PermutationGroup(v) sage: H = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_isomorphic(H) False sage: G.is_isomorphic(G) True sage: G.is_isomorphic(PermutationGroup(list(reversed(v)))) True - >>> from sage.all import * >>> v = ['(1,2,3)(4,5)', '(1,2,3,4,5)'] >>> G = PermutationGroup(v) >>> H = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_isomorphic(H) False >>> G.is_isomorphic(G) True >>> G.is_isomorphic(PermutationGroup(list(reversed(v)))) True 
 - is_monomial()[source]¶
- Return - Trueif the group is monomial. A finite group is monomial if every irreducible complex character is induced from a linear character of a subgroup.- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_monomial() True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_monomial() True 
 - is_nilpotent()[source]¶
- Return - Trueif this group is nilpotent.- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) sage: G.is_nilpotent() False sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_nilpotent() True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) >>> G.is_nilpotent() False >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_nilpotent() True 
 - is_normal(other)[source]¶
- Return - Trueif this group is a normal subgroup of- other.- EXAMPLES: - sage: AlternatingGroup(4).is_normal(SymmetricGroup(4)) True sage: H = PermutationGroup(['(1,2,3)(4,5)']) sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) sage: H.is_normal(G) False - >>> from sage.all import * >>> AlternatingGroup(Integer(4)).is_normal(SymmetricGroup(Integer(4))) True >>> H = PermutationGroup(['(1,2,3)(4,5)']) >>> G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) >>> H.is_normal(G) False 
 - is_perfect()[source]¶
- Return - Trueif this group is perfect. A group is perfect if it equals its derived subgroup.- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) sage: G.is_perfect() False sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_perfect() False - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) >>> G.is_perfect() False >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_perfect() False 
 - is_pgroup()[source]¶
- Return - Trueif this group is a \(p\)-group.- A finite group is a \(p\)-group if its order is of the form \(p^n\) for a prime integer \(p\) and a nonnegative integer \(n\). - EXAMPLES: - sage: G = PermutationGroup(['(1,2,3,4,5)']) sage: G.is_pgroup() True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3,4,5)']) >>> G.is_pgroup() True 
 - is_polycyclic()[source]¶
- Return - Trueif this group is polycyclic. A group is polycyclic if it has a subnormal series with cyclic factors. (For finite groups, this is the same as if the group is solvable - see- is_solvable().)- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) sage: G.is_polycyclic() False sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_polycyclic() True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) >>> G.is_polycyclic() False >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_polycyclic() True 
 - is_primitive(domain=None)[source]¶
- Return - Trueif- selfacts primitively on- domain.- A group \(G\) acts primitively on a set \(S\) if - \(G\) acts transitively on \(S\) and 
- the action induces no non-trivial block system on \(S\). 
 - INPUT: - domain– optional
 - See also - EXAMPLES: - By default, test for primitivity of - selfon its domain:- sage: G = PermutationGroup([[(1,2,3,4)],[(1,2)]]) sage: G.is_primitive() True sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]]) sage: G.is_primitive() False - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(1),Integer(2))]]) >>> G.is_primitive() True >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(2),Integer(4))]]) >>> G.is_primitive() False - You can specify a domain on which to test primitivity: - sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]]) sage: G.is_primitive([1..4]) False sage: G = PermutationGroup([[(3,4,5,6)],[(3,4)]]) #S_4 on [3..6] sage: G.is_primitive(G.non_fixed_points()) True - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(2),Integer(4))]]) >>> G.is_primitive((ellipsis_range(Integer(1),Ellipsis,Integer(4)))) False >>> G = PermutationGroup([[(Integer(3),Integer(4),Integer(5),Integer(6))],[(Integer(3),Integer(4))]]) #S_4 on [3..6] >>> G.is_primitive(G.non_fixed_points()) True - If \(G\) does not act on the domain, it always returns - False:- sage: G = PermutationGroup([[(1,2,3,4)],[(2,4)]]) sage: G.is_primitive([1,2,3]) False - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(2),Integer(4))]]) >>> G.is_primitive([Integer(1),Integer(2),Integer(3)]) False 
 - is_regular(domain=None)[source]¶
- Return - Trueif- selfacts regularly on- domain.- A group \(G\) acts regularly on a set \(S\) if - \(G\) acts transitively on \(S\) and 
- \(G\) acts semi-regularly on \(S\). 
 - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3,4)]]) sage: G.is_regular() True sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]]) sage: G.is_regular() False - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> G.is_regular() True >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(5),Integer(6))]]) >>> G.is_regular() False - You can pass in a domain on which to test regularity: - sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]]) sage: G.is_regular([1..4]) True sage: G.is_regular(G.non_fixed_points()) False - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(5),Integer(6))]]) >>> G.is_regular((ellipsis_range(Integer(1),Ellipsis,Integer(4)))) True >>> G.is_regular(G.non_fixed_points()) False 
 - is_semi_regular(domain=None)[source]¶
- Return - Trueif- selfacts semi-regularly on- domain.- A group \(G\) acts semi-regularly on a set \(S\) if the point stabilizers of \(S\) in \(G\) are trivial. - domainis optional and may take several forms. See examples.- EXAMPLES: - sage: G = PermutationGroup([[(1,2,3,4)]]) sage: G.is_semi_regular() True sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]]) sage: G.is_semi_regular() False - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> G.is_semi_regular() True >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(5),Integer(6))]]) >>> G.is_semi_regular() False - You can pass in a domain to test semi-regularity: - sage: G = PermutationGroup([[(1,2,3,4)],[(5,6)]]) sage: G.is_semi_regular([1..4]) True sage: G.is_semi_regular(G.non_fixed_points()) False - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(5),Integer(6))]]) >>> G.is_semi_regular((ellipsis_range(Integer(1),Ellipsis,Integer(4)))) True >>> G.is_semi_regular(G.non_fixed_points()) False 
 - is_simple()[source]¶
- Return - Trueif the group is simple.- A group is simple if it has no proper normal subgroups. - EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_simple() False - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_simple() False 
 - is_solvable()[source]¶
- Return - Trueif the group is solvable.- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_solvable() True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_solvable() True 
 - is_subgroup(other)[source]¶
- Return - Trueif- selfis a subgroup of- other.- EXAMPLES: - sage: G = AlternatingGroup(5) sage: H = SymmetricGroup(5) sage: G.is_subgroup(H) True - >>> from sage.all import * >>> G = AlternatingGroup(Integer(5)) >>> H = SymmetricGroup(Integer(5)) >>> G.is_subgroup(H) True 
 - is_supersolvable()[source]¶
- Return - Trueif the group is supersolvable.- A finite group is supersolvable if it has a normal series with cyclic factors. - EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: G.is_supersolvable() True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> G.is_supersolvable() True 
 - is_transitive(domain=None)[source]¶
- Return - Trueif- selfacts transitively on- domain.- A group \(G\) acts transitively on set \(S\) if for all \(x,y\in S\) there is some \(g\in G\) such that \(x^g=y\). - EXAMPLES: - sage: G = SymmetricGroup(5) sage: G.is_transitive() True sage: G = PermutationGroup(['(1,2)(3,4)(5,6)']) sage: G.is_transitive() False - >>> from sage.all import * >>> G = SymmetricGroup(Integer(5)) >>> G.is_transitive() True >>> G = PermutationGroup(['(1,2)(3,4)(5,6)']) >>> G.is_transitive() False - sage: G = PermutationGroup([[(1,2,3,4,5)],[(1,2)],[(6,7)]]) sage: G.is_transitive([1,2,3,4,5]) True sage: G.is_transitive([1..7]) False sage: G.is_transitive(G.non_fixed_points()) False sage: H = PermutationGroup([[(1,2,3)],[(4,5,6)]]) sage: H.is_transitive(H.non_fixed_points()) False - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4),Integer(5))],[(Integer(1),Integer(2))],[(Integer(6),Integer(7))]]) >>> G.is_transitive([Integer(1),Integer(2),Integer(3),Integer(4),Integer(5)]) True >>> G.is_transitive((ellipsis_range(Integer(1),Ellipsis,Integer(7)))) False >>> G.is_transitive(G.non_fixed_points()) False >>> H = PermutationGroup([[(Integer(1),Integer(2),Integer(3))],[(Integer(4),Integer(5),Integer(6))]]) >>> H.is_transitive(H.non_fixed_points()) False - If \(G\) does not act on the domain, it always returns - False:- sage: G = PermutationGroup([[(1,2,3,4,5)],[(1,2)]]) #S_5 on [1..5] sage: G.is_transitive([1,4,5]) False - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4),Integer(5))],[(Integer(1),Integer(2))]]) #S_5 on [1..5] >>> G.is_transitive([Integer(1),Integer(4),Integer(5)]) False - Note that this differs from the definition in GAP, where - IsTransitivereturns whether the group is transitive on the set of points moved by the group.- sage: G = PermutationGroup([(2,3)]) sage: G.is_transitive() False sage: gap(G).IsTransitive() true - >>> from sage.all import * >>> G = PermutationGroup([(Integer(2),Integer(3))]) >>> G.is_transitive() False >>> gap(G).IsTransitive() true 
 - is_trivial()[source]¶
- Return - Trueif this group is the trivial group.- A permutation group is trivial, if it consists only of the identity element, that is, if it has no generators or only trivial generators. - EXAMPLES: - sage: G = PermutationGroup([], domain=["a", "b", "c"]) sage: G.is_trivial() True sage: SymmetricGroup(0).is_trivial() True sage: SymmetricGroup(1).is_trivial() True sage: SymmetricGroup(2).is_trivial() False sage: DihedralGroup(1).is_trivial() False - >>> from sage.all import * >>> G = PermutationGroup([], domain=["a", "b", "c"]) >>> G.is_trivial() True >>> SymmetricGroup(Integer(0)).is_trivial() True >>> SymmetricGroup(Integer(1)).is_trivial() True >>> SymmetricGroup(Integer(2)).is_trivial() False >>> DihedralGroup(Integer(1)).is_trivial() False 
 - isomorphism_to(right)[source]¶
- Return an isomorphism from - selfto- rightif the groups are isomorphic, otherwise- None.- INPUT: - self– this group
- right– a permutation group
 - OUTPUT: - None, or a morphism of permutation groups- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) sage: H = PermutationGroup(['(1,2,3)(4,5)']) sage: G.isomorphism_to(H) is None True sage: G = PermutationGroup([(1,2,3), (2,3)]) sage: H = PermutationGroup([(1,2,4), (1,4)]) sage: G.isomorphism_to(H) # not tested, see below Permutation group morphism: From: Permutation Group with generators [(2,3), (1,2,3)] To: Permutation Group with generators [(1,2,4), (1,4)] Defn: [(2,3), (1,2,3)] -> [(2,4), (1,2,4)] - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) >>> H = PermutationGroup(['(1,2,3)(4,5)']) >>> G.isomorphism_to(H) is None True >>> G = PermutationGroup([(Integer(1),Integer(2),Integer(3)), (Integer(2),Integer(3))]) >>> H = PermutationGroup([(Integer(1),Integer(2),Integer(4)), (Integer(1),Integer(4))]) >>> G.isomorphism_to(H) # not tested, see below Permutation group morphism: From: Permutation Group with generators [(2,3), (1,2,3)] To: Permutation Group with generators [(1,2,4), (1,4)] Defn: [(2,3), (1,2,3)] -> [(2,4), (1,2,4)] 
 - isomorphism_type_info_simple_group()[source]¶
- If the group is simple, then this returns the name of the group. - EXAMPLES: - sage: G = CyclicPermutationGroup(5) sage: G.isomorphism_type_info_simple_group() rec( name := "Z(5)", parameter := 5, series := "Z", shortname := "C5" ) - >>> from sage.all import * >>> G = CyclicPermutationGroup(Integer(5)) >>> G.isomorphism_type_info_simple_group() rec( name := "Z(5)", parameter := 5, series := "Z", shortname := "C5" ) 
 - iteration(algorithm='SGS')[source]¶
- Return an iterator over the elements of this group. - INPUT: - algorithm– (default:- 'SGS') either- 'SGS'– using strong generating system
- 'BFS'– a breadth first search on the Cayley graph with respect to- self.gens()
- 'DFS'– a depth first search on the Cayley graph with respect to- self.gens()
 
 - Note - In general, the algorithm - 'SGS'is faster. Yet, for small groups,- 'BFS'and- 'DFS'might be faster.- Note - The order in which the iterator visits the elements differs in the algorithms. - EXAMPLES: - sage: G = PermutationGroup([[(1,2)], [(2,3)]]) sage: list(G.iteration()) [(), (1,2,3), (1,3,2), (2,3), (1,2), (1,3)] sage: list(G.iteration(algorithm='BFS')) [(), (2,3), (1,2), (1,2,3), (1,3,2), (1,3)] sage: list(G.iteration(algorithm='DFS')) [(), (1,2), (1,3,2), (1,3), (1,2,3), (2,3)] - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2))], [(Integer(2),Integer(3))]]) >>> list(G.iteration()) [(), (1,2,3), (1,3,2), (2,3), (1,2), (1,3)] >>> list(G.iteration(algorithm='BFS')) [(), (2,3), (1,2), (1,2,3), (1,3,2), (1,3)] >>> list(G.iteration(algorithm='DFS')) [(), (1,2), (1,3,2), (1,3), (1,2,3), (2,3)] 
 - largest_moved_point()[source]¶
- Return the largest point moved by a permutation in this group. - EXAMPLES: - sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) sage: G.largest_moved_point() 4 sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.largest_moved_point() 10 - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> G.largest_moved_point() 4 >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4),Integer(10))]]) >>> G.largest_moved_point() 10 - sage: G = PermutationGroup([[('a','b','c'),('d','e')]]) sage: G.largest_moved_point() 'e' - >>> from sage.all import * >>> G = PermutationGroup([[('a','b','c'),('d','e')]]) >>> G.largest_moved_point() 'e' - Warning - The name of this function is not good; this function should be deprecated in term of degree: - sage: P = PermutationGroup([[1,2,3,4]]) sage: P.largest_moved_point() 4 sage: P.cardinality() 1 - >>> from sage.all import * >>> P = PermutationGroup([[Integer(1),Integer(2),Integer(3),Integer(4)]]) >>> P.largest_moved_point() 4 >>> P.cardinality() 1 
 - list()[source]¶
- Return list of all elements of this group. - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]]) sage: G.list() [(), (1,4)(2,3), (1,2)(3,4), (1,3)(2,4), (2,4,3), (1,4,2), (1,2,3), (1,3,4), (2,3,4), (1,4,3), (1,2,4), (1,3,2), (3,4), (1,4,2,3), (1,2), (1,3,2,4), (2,4), (1,4,3,2), (1,2,3,4), (1,3), (2,3), (1,4), (1,2,4,3), (1,3,4,2)] sage: G = PermutationGroup([[('a','b')]], domain=('a', 'b')); G Permutation Group with generators [('a','b')] sage: G.list() [(), ('a','b')] - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))], [(Integer(1),Integer(2))]]) >>> G.list() [(), (1,4)(2,3), (1,2)(3,4), (1,3)(2,4), (2,4,3), (1,4,2), (1,2,3), (1,3,4), (2,3,4), (1,4,3), (1,2,4), (1,3,2), (3,4), (1,4,2,3), (1,2), (1,3,2,4), (2,4), (1,4,3,2), (1,2,3,4), (1,3), (2,3), (1,4), (1,2,4,3), (1,3,4,2)] >>> G = PermutationGroup([[('a','b')]], domain=('a', 'b')); G Permutation Group with generators [('a','b')] >>> G.list() [(), ('a','b')] 
 - lower_central_series()[source]¶
- Return the lower central series of this group as a list of permutation groups. - EXAMPLES: - These computations use pseudo-random numbers, so we set the seed for reproducible testing. - sage: set_random_seed(0) sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: G.lower_central_series() [Subgroup generated by [(3,4), (1,2,3)(4,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]), Subgroup generated by [(1,5)(3,4), (1,5)(2,4), (1,3,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])] - >>> from sage.all import * >>> set_random_seed(Integer(0)) >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]]) >>> G.lower_central_series() [Subgroup generated by [(3,4), (1,2,3)(4,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)]), Subgroup generated by [(1,5)(3,4), (1,5)(2,4), (1,3,5)] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])] 
 - maximal_normal_subgroups()[source]¶
- Return the maximal proper normal subgroups of - self.- This raises an error if \(G/[G, G]\) is infinite, yielding infinitely many maximal normal subgroups. - EXAMPLES: - sage: G = PermutationGroup([(1,2,3),(4,5)]) sage: G.maximal_normal_subgroups() [Subgroup generated by [(1,2,3)] of (Permutation Group with generators [(4,5), (1,2,3)]), Subgroup generated by [(4,5)] of (Permutation Group with generators [(4,5), (1,2,3)])] - >>> from sage.all import * >>> G = PermutationGroup([(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))]) >>> G.maximal_normal_subgroups() [Subgroup generated by [(1,2,3)] of (Permutation Group with generators [(4,5), (1,2,3)]), Subgroup generated by [(4,5)] of (Permutation Group with generators [(4,5), (1,2,3)])] 
 - minimal_generating_set()[source]¶
- Return a minimal generating set. - EXAMPLES: - sage: # needs sage.graphs sage: g = graphs.CompleteGraph(4) sage: g.relabel(['a','b','c','d']) sage: mgs = g.automorphism_group().minimal_generating_set(); len(mgs) 2 sage: mgs # random [('b','d','c'), ('a','c','b','d')] - >>> from sage.all import * >>> # needs sage.graphs >>> g = graphs.CompleteGraph(Integer(4)) >>> g.relabel(['a','b','c','d']) >>> mgs = g.automorphism_group().minimal_generating_set(); len(mgs) 2 >>> mgs # random [('b','d','c'), ('a','c','b','d')] 
 - minimal_normal_subgroups()[source]¶
- Return the nontrivial minimal normal subgroups - self.- EXAMPLES: - sage: G = PermutationGroup([(1,2,3),(4,5)]) sage: G.minimal_normal_subgroups() [Subgroup generated by [(4,5)] of (Permutation Group with generators [(4,5), (1,2,3)]), Subgroup generated by [(1,2,3)] of (Permutation Group with generators [(4,5), (1,2,3)])] - >>> from sage.all import * >>> G = PermutationGroup([(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))]) >>> G.minimal_normal_subgroups() [Subgroup generated by [(4,5)] of (Permutation Group with generators [(4,5), (1,2,3)]), Subgroup generated by [(1,2,3)] of (Permutation Group with generators [(4,5), (1,2,3)])] 
 - molien_series()[source]¶
- Return the Molien series of a permutation group. The function \[M(x) = (1/|G|)\sum_{g\in G} \det(1-x*g)^{-1}\]- is sometimes called the “Molien series” of \(G\). GAP’s - MolienSeriesis associated to a character of a group \(G\). How are these related? A group \(G\), given as a permutation group on \(n\) points, has a “natural” representation of dimension \(n\), given by permutation matrices. The Molien series of \(G\) is the one associated to that permutation representation of \(G\) using the above formula. Character values then count fixed points of the corresponding permutations.- EXAMPLES: - sage: G = SymmetricGroup(5) sage: G.molien_series() -1/(x^15 - x^14 - x^13 + x^10 + x^9 + x^8 - x^7 - x^6 - x^5 + x^2 + x - 1) sage: G = SymmetricGroup(3) sage: G.molien_series() -1/(x^6 - x^5 - x^4 + x^2 + x - 1) - >>> from sage.all import * >>> G = SymmetricGroup(Integer(5)) >>> G.molien_series() -1/(x^15 - x^14 - x^13 + x^10 + x^9 + x^8 - x^7 - x^6 - x^5 + x^2 + x - 1) >>> G = SymmetricGroup(Integer(3)) >>> G.molien_series() -1/(x^6 - x^5 - x^4 + x^2 + x - 1) - Some further tests (after Issue #15817): - sage: G = PermutationGroup([[(1,2,3,4)]]) sage: S4ms = SymmetricGroup(4).molien_series() sage: G.molien_series() / S4ms x^5 + 2*x^4 + x^3 + x^2 + 1 - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> S4ms = SymmetricGroup(Integer(4)).molien_series() >>> G.molien_series() / S4ms x^5 + 2*x^4 + x^3 + x^2 + 1 - This works for not-transitive groups: - sage: G = PermutationGroup([[(1,2)],[(3,4)]]) sage: G.molien_series() / S4ms x^4 + x^3 + 2*x^2 + x + 1 - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2))],[(Integer(3),Integer(4))]]) >>> G.molien_series() / S4ms x^4 + x^3 + 2*x^2 + x + 1 - This works for groups with fixed points: - sage: G = PermutationGroup([[(2,)]]) sage: G.molien_series() 1/(x^2 - 2*x + 1) - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(2),)]]) >>> G.molien_series() 1/(x^2 - 2*x + 1) 
 - ngens()[source]¶
- Return the number of generators of - self.- EXAMPLES: - sage: A4 = PermutationGroup([[(1,2,3)], [(2,3,4)]]); A4 Permutation Group with generators [(2,3,4), (1,2,3)] sage: A4.ngens() 2 - >>> from sage.all import * >>> A4 = PermutationGroup([[(Integer(1),Integer(2),Integer(3))], [(Integer(2),Integer(3),Integer(4))]]); A4 Permutation Group with generators [(2,3,4), (1,2,3)] >>> A4.ngens() 2 
 - non_fixed_points()[source]¶
- Return the list of points not fixed by - self, i.e., the subset of- self.domain()moved by some element of- self.- EXAMPLES: - sage: G = PermutationGroup([[(3,4,5)],[(7,10)]]) sage: G.non_fixed_points() [3, 4, 5, 7, 10] sage: G = PermutationGroup([[(2,3,6)],[(9,)]]) # note: 9 is fixed sage: G.non_fixed_points() [2, 3, 6] - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(3),Integer(4),Integer(5))],[(Integer(7),Integer(10))]]) >>> G.non_fixed_points() [3, 4, 5, 7, 10] >>> G = PermutationGroup([[(Integer(2),Integer(3),Integer(6))],[(Integer(9),)]]) # note: 9 is fixed >>> G.non_fixed_points() [2, 3, 6] 
 - normal_subgroups()[source]¶
- Return the normal subgroups of this group as a (sorted in increasing order) list of permutation groups. - The normal subgroups of \(H = PSL(2,7) \times PSL(2,7)\) are \(1\), two copies of \(PSL(2,7)\) and \(H\) itself, as the following example shows. - EXAMPLES: - sage: G = PSL(2,7) sage: D = G.direct_product(G) sage: H = D[0] sage: NH = H.normal_subgroups() sage: len(NH) 4 sage: NH[1].is_isomorphic(G) True sage: NH[2].is_isomorphic(G) True - >>> from sage.all import * >>> G = PSL(Integer(2),Integer(7)) >>> D = G.direct_product(G) >>> H = D[Integer(0)] >>> NH = H.normal_subgroups() >>> len(NH) 4 >>> NH[Integer(1)].is_isomorphic(G) True >>> NH[Integer(2)].is_isomorphic(G) True 
 - normalizer(g)[source]¶
- Return the normalizer of - gin- self.- EXAMPLES: - sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]]) sage: g = G([(1,3)]) sage: G.normalizer(g) Subgroup generated by [(1,3), (2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) sage: g = G([(1,2,3,4)]) sage: G.normalizer(g) Subgroup generated by [(1,2,3,4), (1,3)(2,4), (2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) sage: H = G.subgroup([G([(1,2,3,4)])]) sage: G.normalizer(H) Subgroup generated by [(1,2,3,4), (1,3)(2,4), (2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4))]]) >>> g = G([(Integer(1),Integer(3))]) >>> G.normalizer(g) Subgroup generated by [(1,3), (2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) >>> g = G([(Integer(1),Integer(2),Integer(3),Integer(4))]) >>> G.normalizer(g) Subgroup generated by [(1,2,3,4), (1,3)(2,4), (2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) >>> H = G.subgroup([G([(Integer(1),Integer(2),Integer(3),Integer(4))])]) >>> G.normalizer(H) Subgroup generated by [(1,2,3,4), (1,3)(2,4), (2,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4)]) 
 - normalizes(other)[source]¶
- Return - Trueif the group- otheris normalized by- self.- Wraps GAP’s - IsNormalfunction.- A group \(G\) normalizes a group \(U\) if and only if for every \(g \in G\) and \(u \in U\) the element \(u^g\) is a member of \(U\). Note that \(U\) need not be a subgroup of \(G\). - EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)(4,5)']) sage: H = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) sage: H.normalizes(G) False sage: G = SymmetricGroup(3) sage: H = PermutationGroup( [ (4,5,6) ] ) sage: G.normalizes(H) True sage: H.normalizes(G) True - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)(4,5)']) >>> H = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)']) >>> H.normalizes(G) False >>> G = SymmetricGroup(Integer(3)) >>> H = PermutationGroup( [ (Integer(4),Integer(5),Integer(6)) ] ) >>> G.normalizes(H) True >>> H.normalizes(G) True - In the last example, \(G\) and \(H\) are disjoint, so each normalizes the other. 
 - one()[source]¶
- Return the identity element of this group. - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3),(4,5)]]) sage: e = G.identity(); e # indirect doctest () sage: g = G.gen(0) sage: g*e (1,2,3)(4,5) sage: e*g (1,2,3)(4,5) sage: S = SymmetricGroup(['a','b','c']) sage: S.identity() () - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))]]) >>> e = G.identity(); e # indirect doctest () >>> g = G.gen(Integer(0)) >>> g*e (1,2,3)(4,5) >>> e*g (1,2,3)(4,5) >>> S = SymmetricGroup(['a','b','c']) >>> S.identity() () 
 - orbit(point, action='OnPoints')[source]¶
- Return the orbit of a point under a group action. - INPUT: - point– can be a point or any of the list above, depending on the action to be considered
- action– string; if- pointis an element from the domain, a tuple of elements of the domain, a tuple of tuples […], this variable describes how the group is acting- The actions currently available through this method are - 'OnPoints',- 'OnTuples',- 'OnSets',- 'OnPairs',- 'OnSetsSets',- 'OnSetsDisjointSets',- 'OnSetsTuples',- 'OnTuplesSets',- 'OnTuplesTuples'. They are taken from GAP’s list of group actions, see- gap.help('Group Actions').- It is set to - 'OnPoints'by default. See below for examples.
 - OUTPUT: - The orbit of - pointas a tuple. Each entry is an image under the action of the permutation group, if necessary converted to the corresponding container. That is, if- action='OnSets'then each entry will be a set even if- pointwas given by a list/tuple/iterable.- EXAMPLES: - sage: G = PermutationGroup([ [(3,4)], [(1,3)] ]) sage: G.orbit(3) (3, 4, 1) sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.orbit(3) (3, 4, 10, 1, 2) sage: G = PermutationGroup([ [('c','d')], [('a','c')] ]) sage: G.orbit('a') ('a', 'c', 'd') - >>> from sage.all import * >>> G = PermutationGroup([ [(Integer(3),Integer(4))], [(Integer(1),Integer(3))] ]) >>> G.orbit(Integer(3)) (3, 4, 1) >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4),Integer(10))]]) >>> G.orbit(Integer(3)) (3, 4, 10, 1, 2) >>> G = PermutationGroup([ [('c','d')], [('a','c')] ]) >>> G.orbit('a') ('a', 'c', 'd') - Action of \(S_3\) on sets: - sage: S3 = groups.permutation.Symmetric(3) sage: S3.orbit((1,2), action = "OnSets") ({1, 2}, {2, 3}, {1, 3}) - >>> from sage.all import * >>> S3 = groups.permutation.Symmetric(Integer(3)) >>> S3.orbit((Integer(1),Integer(2)), action = "OnSets") ({1, 2}, {2, 3}, {1, 3}) - On tuples: - sage: S3.orbit((1,2), action = "OnTuples") ((1, 2), (2, 3), (2, 1), (3, 1), (1, 3), (3, 2)) - >>> from sage.all import * >>> S3.orbit((Integer(1),Integer(2)), action = "OnTuples") ((1, 2), (2, 3), (2, 1), (3, 1), (1, 3), (3, 2)) - Action of \(S_4\) on sets of disjoint sets: - sage: S4 = groups.permutation.Symmetric(4) sage: O = S4.orbit(((1,2),(3,4)), action='OnSetsDisjointSets') sage: {1, 2} in O[0] and {3, 4} in O[0] True sage: {1, 4} in O[1] and {2, 3} in O[1] True sage: all(x[0].union(x[1]) == {1,2,3,4} for x in O) True - >>> from sage.all import * >>> S4 = groups.permutation.Symmetric(Integer(4)) >>> O = S4.orbit(((Integer(1),Integer(2)),(Integer(3),Integer(4))), action='OnSetsDisjointSets') >>> {Integer(1), Integer(2)} in O[Integer(0)] and {Integer(3), Integer(4)} in O[Integer(0)] True >>> {Integer(1), Integer(4)} in O[Integer(1)] and {Integer(2), Integer(3)} in O[Integer(1)] True >>> all(x[Integer(0)].union(x[Integer(1)]) == {Integer(1),Integer(2),Integer(3),Integer(4)} for x in O) True - Action of \(S_4\) (on a nonstandard domain) on tuples of sets: - sage: S4 = PermutationGroup([ [('c','d')], [('a','c')], [('a','b')] ]) sage: orb = S4.orbit((('a','c'),('b','d')), "OnTuplesSets") sage: expect = (({'a', 'c'}, {'b', 'd'}), ({'a', 'd'}, {'c', 'b'}), ....: ({'c', 'b'}, {'a', 'd'}), ({'b', 'd'}, {'a', 'c'}), ....: ({'c', 'd'}, {'a', 'b'}), ({'a', 'b'}, {'c', 'd'})) sage: expect == orb True - >>> from sage.all import * >>> S4 = PermutationGroup([ [('c','d')], [('a','c')], [('a','b')] ]) >>> orb = S4.orbit((('a','c'),('b','d')), "OnTuplesSets") >>> expect = (({'a', 'c'}, {'b', 'd'}), ({'a', 'd'}, {'c', 'b'}), ... ({'c', 'b'}, {'a', 'd'}), ({'b', 'd'}, {'a', 'c'}), ... ({'c', 'd'}, {'a', 'b'}), ({'a', 'b'}, {'c', 'd'})) >>> expect == orb True - Action of \(S_4\) (on a very nonstandard domain) on tuples of sets: - sage: S4 = PermutationGroup([ [((11,(12,13)),'d')], ....: [((12,(12,11)),(11,(12,13)))], [((12,(12,11)),'b')] ]) sage: orb = S4.orbit((( (11,(12,13)), (12,(12,11))),('b','d')), ....: "OnTuplesSets") sage: expect = (({(11, (12, 13)), (12, (12, 11))}, {'b', 'd'}), ....: ({'d', (12, (12, 11))}, {(11, (12, 13)), 'b'}), ....: ({(11, (12, 13)), 'b'}, {'d', (12, (12, 11))}), ....: ({(11, (12, 13)), 'd'}, {'b', (12, (12, 11))}), ....: ({'b', 'd'}, {(11, (12, 13)), (12, (12, 11))}), ....: ({'b', (12, (12, 11))}, {(11, (12, 13)), 'd'})) sage: all(x in orb for x in expect) and len(orb) == len(expect) True - >>> from sage.all import * >>> S4 = PermutationGroup([ [((Integer(11),(Integer(12),Integer(13))),'d')], ... [((Integer(12),(Integer(12),Integer(11))),(Integer(11),(Integer(12),Integer(13))))], [((Integer(12),(Integer(12),Integer(11))),'b')] ]) >>> orb = S4.orbit((( (Integer(11),(Integer(12),Integer(13))), (Integer(12),(Integer(12),Integer(11)))),('b','d')), ... "OnTuplesSets") >>> expect = (({(Integer(11), (Integer(12), Integer(13))), (Integer(12), (Integer(12), Integer(11)))}, {'b', 'd'}), ... ({'d', (Integer(12), (Integer(12), Integer(11)))}, {(Integer(11), (Integer(12), Integer(13))), 'b'}), ... ({(Integer(11), (Integer(12), Integer(13))), 'b'}, {'d', (Integer(12), (Integer(12), Integer(11)))}), ... ({(Integer(11), (Integer(12), Integer(13))), 'd'}, {'b', (Integer(12), (Integer(12), Integer(11)))}), ... ({'b', 'd'}, {(Integer(11), (Integer(12), Integer(13))), (Integer(12), (Integer(12), Integer(11)))}), ... ({'b', (Integer(12), (Integer(12), Integer(11)))}, {(Integer(11), (Integer(12), Integer(13))), 'd'})) >>> all(x in orb for x in expect) and len(orb) == len(expect) True 
 - orbits()[source]¶
- Return the orbits of the elements of the domain under the default group action. - EXAMPLES: - sage: G = PermutationGroup([ [(3,4)], [(1,3)] ]) sage: G.orbits() ((1, 3, 4), (2,)) sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.orbits() ((1, 2, 3, 4, 10), (5,), (6,), (7,), (8,), (9,)) sage: G = PermutationGroup([ [('c','d')], [('a','c')],[('b',)]]) sage: G.orbits() (('a', 'c', 'd'), ('b',)) - >>> from sage.all import * >>> G = PermutationGroup([ [(Integer(3),Integer(4))], [(Integer(1),Integer(3))] ]) >>> G.orbits() ((1, 3, 4), (2,)) >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4),Integer(10))]]) >>> G.orbits() ((1, 2, 3, 4, 10), (5,), (6,), (7,), (8,), (9,)) >>> G = PermutationGroup([ [('c','d')], [('a','c')],[('b',)]]) >>> G.orbits() (('a', 'c', 'd'), ('b',)) - The answer is cached: - sage: G.orbits() is G.orbits() True - >>> from sage.all import * >>> G.orbits() is G.orbits() True - AUTHORS: - Nathan Dunfield 
 
 - order()[source]¶
- Return the number of elements of this group. - See also: - degree().- EXAMPLES: - sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]]) sage: G.order() 12 sage: G = PermutationGroup([()]) sage: G.order() 1 sage: G = PermutationGroup([]) sage: G.order() 1 - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))], [(Integer(1),Integer(2))]]) >>> G.order() 12 >>> G = PermutationGroup([()]) >>> G.order() 1 >>> G = PermutationGroup([]) >>> G.order() 1 - cardinality()is just an alias:- sage: PermutationGroup([(1,2,3)]).cardinality() 3 - >>> from sage.all import * >>> PermutationGroup([(Integer(1),Integer(2),Integer(3))]).cardinality() 3 
 - poincare_series(p=2, n=10)[source]¶
- Return the Poincaré series of \(G \mod p\) (\(p \geq 2\) must be a prime), for \(n\) large. - In other words, if you input a finite group \(G\), a prime \(p\), and a positive integer \(n\), it returns a quotient of polynomials \(f(x) = P(x) / Q(x)\) whose coefficient of \(x^k\) equals the rank of the vector space \(H_k(G, \ZZ / p \ZZ)\), for all \(k\) in the range \(1 \leq k \leq n\). - REQUIRES: GAP package HAP (in gap_packages-*.spkg). - EXAMPLES: - sage: G = SymmetricGroup(5) sage: G.poincare_series(2, 10) # optional - gap_package_hap (x^2 + 1)/(x^4 - x^3 - x + 1) sage: G = SymmetricGroup(3) sage: G.poincare_series(2, 10) # optional - gap_package_hap -1/(x - 1) - >>> from sage.all import * >>> G = SymmetricGroup(Integer(5)) >>> G.poincare_series(Integer(2), Integer(10)) # optional - gap_package_hap (x^2 + 1)/(x^4 - x^3 - x + 1) >>> G = SymmetricGroup(Integer(3)) >>> G.poincare_series(Integer(2), Integer(10)) # optional - gap_package_hap -1/(x - 1) - AUTHORS: - David Joyner and Graham Ellis 
 
 - quotient(N, **kwds)[source]¶
- Return the quotient of this permutation group by the normal subgroup \(N\), as a permutation group. - Further named arguments are passed to the permutation group constructor. - Wraps the GAP operator “/”. - EXAMPLES: - sage: G = PermutationGroup([(1,2,3), (2,3)]) sage: N = PermutationGroup([(1,2,3)]) sage: G.quotient(N) Permutation Group with generators [(1,2)] sage: G.quotient(G) Permutation Group with generators [(), ()] - >>> from sage.all import * >>> G = PermutationGroup([(Integer(1),Integer(2),Integer(3)), (Integer(2),Integer(3))]) >>> N = PermutationGroup([(Integer(1),Integer(2),Integer(3))]) >>> G.quotient(N) Permutation Group with generators [(1,2)] >>> G.quotient(G) Permutation Group with generators [(), ()] 
 - random_element()[source]¶
- Return a random element of this group. - EXAMPLES: - sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]]) sage: a = G.random_element() sage: a in G True sage: a.parent() is G True sage: a^6 () - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))], [(Integer(1),Integer(2))]]) >>> a = G.random_element() >>> a in G True >>> a.parent() is G True >>> a**Integer(6) () 
 - representative_action(x, y)[source]¶
- Return an element of - selfthat maps \(x\) to \(y\) if it exists.- This method wraps the gap function - RepresentativeAction, which can also return elements that map a given set of points on another set of points.- INPUT: - x,- y– two elements of the domain
 - EXAMPLES: - sage: G = groups.permutation.Cyclic(14) sage: g = G.representative_action(1, 10) sage: all(g(x) == 1 + ((x+9-1)%14) for x in G.domain()) True - >>> from sage.all import * >>> G = groups.permutation.Cyclic(Integer(14)) >>> g = G.representative_action(Integer(1), Integer(10)) >>> all(g(x) == Integer(1) + ((x+Integer(9)-Integer(1))%Integer(14)) for x in G.domain()) True 
 - semidirect_product(N, mapping, check=True)[source]¶
- The semidirect product of - selfwith- N.- INPUT: - N– a group which is acted on by- selfand naturally embeds as a normal subgroup of the returned semidirect product
- mapping– a pair of lists that together define a homomorphism, \(\phi :\)- self\(\rightarrow\) Aut(- N), by giving, in the second list, the images of the generators of- selfin the order given in the first list
- check– a boolean that, if set to- False, will skip the initial tests which are made on- mapping. This may be beneficial for large- N, since in such cases the injectivity test can be expensive. Set to- Trueby default.
 - OUTPUT: - The semidirect product of - selfand- Ndefined by the action of- selfon- Ngiven in- mapping(note that a homomorphism from \(A\) to the automorphism group of \(B\) is equivalent to an action of \(A\) on \(B\)’s underlying set). The semidirect product of two groups, \(H\) and \(N\), is a construct similar to the direct product in so far as the elements are the Cartesian product of the elements of \(H\) and the elements of \(N\). The operation, however, is built upon an action of \(H\) on \(N\), and is defined as such:\[(h_1,n_1)(h_2,n_2) = (h_{1}h_{2}, n_{1}^{h_2}n_2)\]- This function is a wrapper for GAP’s - SemidirectProductcommand. The permutation group returned is built upon a permutation representation of the semidirect product of- selfand- Non a set of size \(\mid N \mid\). The generators of- Nare given as their right regular representations, while the generators of- selfare defined by the underlying action of- selfon- N. It should be noted that the defining action is not always faithful, and in this case the inputted representations of the generators of- selfare placed on additional letters and adjoined to the output’s generators of- self.- EXAMPLES: - Perhaps the most common example of a semidirect product comes from the family of dihedral groups. Each dihedral group is the semidirect product of \(C_2\) with \(C_n\), where, by convention, \(3 \leq n\). In this case, the nontrivial element of \(C_2\) acts on \(C_n\) so as to send each element to its inverse. - sage: C2 = CyclicPermutationGroup(2) sage: C8 = CyclicPermutationGroup(8) sage: alpha = PermutationGroupMorphism_im_gens(C8,C8,[(1,8,7,6,5,4,3,2)]) sage: S = C2.semidirect_product(C8,[[(1,2)],[alpha]]) sage: S == DihedralGroup(8) False sage: S.is_isomorphic(DihedralGroup(8)) True sage: S.gens() ((3,4,5,6,7,8,9,10), (1,2)(4,10)(5,9)(6,8)) - >>> from sage.all import * >>> C2 = CyclicPermutationGroup(Integer(2)) >>> C8 = CyclicPermutationGroup(Integer(8)) >>> alpha = PermutationGroupMorphism_im_gens(C8,C8,[(Integer(1),Integer(8),Integer(7),Integer(6),Integer(5),Integer(4),Integer(3),Integer(2))]) >>> S = C2.semidirect_product(C8,[[(Integer(1),Integer(2))],[alpha]]) >>> S == DihedralGroup(Integer(8)) False >>> S.is_isomorphic(DihedralGroup(Integer(8))) True >>> S.gens() ((3,4,5,6,7,8,9,10), (1,2)(4,10)(5,9)(6,8)) - A more complicated example can be drawn from [TW1980]. It is there given that a semidirect product of \(D_4\) and \(C_3\) is isomorphic to one of \(C_2\) and the dicyclic group of order 12. This nonabelian group of order 24 has very similar structure to the dicyclic and dihedral groups of order 24, the three being the only groups of order 24 with a two-element center and 9 conjugacy classes. - sage: D4 = DihedralGroup(4) sage: C3 = CyclicPermutationGroup(3) sage: alpha1 = PermutationGroupMorphism_im_gens(C3,C3,[(1,3,2)]) sage: alpha2 = PermutationGroupMorphism_im_gens(C3,C3,[(1,2,3)]) sage: S1 = D4.semidirect_product(C3,[[(1,2,3,4),(1,3)],[alpha1,alpha2]]) sage: C2 = CyclicPermutationGroup(2) sage: Q = DiCyclicGroup(3) sage: a = Q.gens()[0]; b=Q.gens()[1].inverse() sage: alpha = PermutationGroupMorphism_im_gens(Q,Q,[a,b]) sage: S2 = C2.semidirect_product(Q,[[(1,2)],[alpha]]) sage: S1.is_isomorphic(S2) True sage: S1.is_isomorphic(DihedralGroup(12)) False sage: S1.is_isomorphic(DiCyclicGroup(6)) False sage: S1.center() Subgroup generated by [(1,3)(2,4)] of (Permutation Group with generators [(5,6,7), (1,2,3,4)(6,7), (1,3)]) sage: len(S1.conjugacy_classes_representatives()) 9 - >>> from sage.all import * >>> D4 = DihedralGroup(Integer(4)) >>> C3 = CyclicPermutationGroup(Integer(3)) >>> alpha1 = PermutationGroupMorphism_im_gens(C3,C3,[(Integer(1),Integer(3),Integer(2))]) >>> alpha2 = PermutationGroupMorphism_im_gens(C3,C3,[(Integer(1),Integer(2),Integer(3))]) >>> S1 = D4.semidirect_product(C3,[[(Integer(1),Integer(2),Integer(3),Integer(4)),(Integer(1),Integer(3))],[alpha1,alpha2]]) >>> C2 = CyclicPermutationGroup(Integer(2)) >>> Q = DiCyclicGroup(Integer(3)) >>> a = Q.gens()[Integer(0)]; b=Q.gens()[Integer(1)].inverse() >>> alpha = PermutationGroupMorphism_im_gens(Q,Q,[a,b]) >>> S2 = C2.semidirect_product(Q,[[(Integer(1),Integer(2))],[alpha]]) >>> S1.is_isomorphic(S2) True >>> S1.is_isomorphic(DihedralGroup(Integer(12))) False >>> S1.is_isomorphic(DiCyclicGroup(Integer(6))) False >>> S1.center() Subgroup generated by [(1,3)(2,4)] of (Permutation Group with generators [(5,6,7), (1,2,3,4)(6,7), (1,3)]) >>> len(S1.conjugacy_classes_representatives()) 9 - If your normal subgroup is large, and you are confident that your inputs will successfully create a semidirect product, then it is beneficial, for the sake of time efficiency, to set the - checkparameter to- False.- sage: C2 = CyclicPermutationGroup(2) sage: C2000 = CyclicPermutationGroup(500) sage: alpha = PermutationGroupMorphism(C2000,C2000,[C2000.gen().inverse()]) sage: S = C2.semidirect_product(C2000,[[(1,2)],[alpha]],check=False) - >>> from sage.all import * >>> C2 = CyclicPermutationGroup(Integer(2)) >>> C2000 = CyclicPermutationGroup(Integer(500)) >>> alpha = PermutationGroupMorphism(C2000,C2000,[C2000.gen().inverse()]) >>> S = C2.semidirect_product(C2000,[[(Integer(1),Integer(2))],[alpha]],check=False) - AUTHOR: - Kevin Halasz (2012-8-12) 
 
 - sign_representation(base_ring=None)[source]¶
- Return the sign representation of - selfover- base_ring.- INPUT: - base_ring– (optional) the base ring; the default is \(\ZZ\)
 - EXAMPLES: - sage: G = groups.permutation.Dihedral(4) sage: G.sign_representation() Sign representation of Dihedral group of order 8 as a permutation group over Integer Ring - >>> from sage.all import * >>> G = groups.permutation.Dihedral(Integer(4)) >>> G.sign_representation() Sign representation of Dihedral group of order 8 as a permutation group over Integer Ring 
 - smallest_moved_point()[source]¶
- Return the smallest point moved by a permutation in this group. - EXAMPLES: - sage: G = PermutationGroup([[(3,4)], [(2,3,4)]]) sage: G.smallest_moved_point() 2 sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.smallest_moved_point() 1 - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(3),Integer(4))], [(Integer(2),Integer(3),Integer(4))]]) >>> G.smallest_moved_point() 2 >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4),Integer(10))]]) >>> G.smallest_moved_point() 1 - Note that this function uses the ordering from the domain: - sage: S = SymmetricGroup(['a','b','c']) sage: S.smallest_moved_point() 'a' - >>> from sage.all import * >>> S = SymmetricGroup(['a','b','c']) >>> S.smallest_moved_point() 'a' 
 - socle()[source]¶
- Return the socle of - self.- The socle of a group \(G\) is the subgroup generated by all minimal normal subgroups. - EXAMPLES: - sage: G = SymmetricGroup(4) sage: s = G.socle(); s Subgroup generated by [(1,2)(3,4), (1,4)(2,3)] of (Symmetric group of order 4! as a permutation group) - >>> from sage.all import * >>> G = SymmetricGroup(Integer(4)) >>> s = G.socle(); s Subgroup generated by [(1,2)(3,4), (1,4)(2,3)] of (Symmetric group of order 4! as a permutation group) - The socle of the socle is, essentially, the socle: - sage: s.socle() == s.subgroup(s.gens()) True - >>> from sage.all import * >>> s.socle() == s.subgroup(s.gens()) True 
 - solvable_radical()[source]¶
- Return the solvable radical of - self.- The solvable radical (or just radical) of a group \(G\) is the largest solvable normal subgroup of \(G\). - EXAMPLES: - sage: G = SymmetricGroup(4) sage: G.solvable_radical() Subgroup generated by [(1,2,3,4), (1,2)] of (Symmetric group of order 4! as a permutation group) sage: G = SymmetricGroup(5) sage: G.solvable_radical() Subgroup generated by [()] of (Symmetric group of order 5! as a permutation group) - >>> from sage.all import * >>> G = SymmetricGroup(Integer(4)) >>> G.solvable_radical() Subgroup generated by [(1,2,3,4), (1,2)] of (Symmetric group of order 4! as a permutation group) >>> G = SymmetricGroup(Integer(5)) >>> G.solvable_radical() Subgroup generated by [()] of (Symmetric group of order 5! as a permutation group) 
 - stabilizer(point, action='OnPoints')[source]¶
- Return the subgroup of - selfwhich stabilize the given position.- selfand its stabilizers must have same degree.- INPUT: - point– a point of the- domain(), or a set of points depending on the value of- action
- action– string (default:- 'OnPoints'); should the group be considered to act on points (- action="OnPoints") or on sets of points (- action="OnSets")? In the latter case, the first argument must be a subset of- domain().
 - EXAMPLES: - sage: G = PermutationGroup([ [(3,4)], [(1,3)] ]) sage: G.stabilizer(1) Subgroup generated by [(3,4)] of (Permutation Group with generators [(3,4), (1,3)]) sage: G.stabilizer(3) Subgroup generated by [(1,4)] of (Permutation Group with generators [(3,4), (1,3)]) - >>> from sage.all import * >>> G = PermutationGroup([ [(Integer(3),Integer(4))], [(Integer(1),Integer(3))] ]) >>> G.stabilizer(Integer(1)) Subgroup generated by [(3,4)] of (Permutation Group with generators [(3,4), (1,3)]) >>> G.stabilizer(Integer(3)) Subgroup generated by [(1,4)] of (Permutation Group with generators [(3,4), (1,3)]) - The stabilizer of a set of points: - sage: s10 = groups.permutation.Symmetric(10) sage: s10.stabilizer([1..3],"OnSets").cardinality() 30240 sage: factorial(3)*factorial(7) 30240 - >>> from sage.all import * >>> s10 = groups.permutation.Symmetric(Integer(10)) >>> s10.stabilizer((ellipsis_range(Integer(1),Ellipsis,Integer(3))),"OnSets").cardinality() 30240 >>> factorial(Integer(3))*factorial(Integer(7)) 30240 - sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.stabilizer(10) Subgroup generated by [(1,2)(3,4), (2,3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4,10)]) sage: G.stabilizer(1) == G.subgroup(['(2,3)(4,10)', '(2,10,3)']) True sage: G = PermutationGroup([[(2,3,4)],[(6,7)]]) sage: G.stabilizer(1) Subgroup generated by [(6,7), (2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)]) sage: G.stabilizer(2) Subgroup generated by [(6,7)] of (Permutation Group with generators [(6,7), (2,3,4)]) sage: G.stabilizer(3) Subgroup generated by [(6,7)] of (Permutation Group with generators [(6,7), (2,3,4)]) sage: G.stabilizer(4) Subgroup generated by [(6,7)] of (Permutation Group with generators [(6,7), (2,3,4)]) sage: G.stabilizer(5) Subgroup generated by [(6,7), (2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)]) sage: G.stabilizer(6) Subgroup generated by [(2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)]) sage: G.stabilizer(7) Subgroup generated by [(2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)]) sage: G.stabilizer(8) Traceback (most recent call last): ... ValueError: 8 does not belong to the domain - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4),Integer(10))]]) >>> G.stabilizer(Integer(10)) Subgroup generated by [(1,2)(3,4), (2,3,4)] of (Permutation Group with generators [(1,2)(3,4), (1,2,3,4,10)]) >>> G.stabilizer(Integer(1)) == G.subgroup(['(2,3)(4,10)', '(2,10,3)']) True >>> G = PermutationGroup([[(Integer(2),Integer(3),Integer(4))],[(Integer(6),Integer(7))]]) >>> G.stabilizer(Integer(1)) Subgroup generated by [(6,7), (2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)]) >>> G.stabilizer(Integer(2)) Subgroup generated by [(6,7)] of (Permutation Group with generators [(6,7), (2,3,4)]) >>> G.stabilizer(Integer(3)) Subgroup generated by [(6,7)] of (Permutation Group with generators [(6,7), (2,3,4)]) >>> G.stabilizer(Integer(4)) Subgroup generated by [(6,7)] of (Permutation Group with generators [(6,7), (2,3,4)]) >>> G.stabilizer(Integer(5)) Subgroup generated by [(6,7), (2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)]) >>> G.stabilizer(Integer(6)) Subgroup generated by [(2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)]) >>> G.stabilizer(Integer(7)) Subgroup generated by [(2,3,4)] of (Permutation Group with generators [(6,7), (2,3,4)]) >>> G.stabilizer(Integer(8)) Traceback (most recent call last): ... ValueError: 8 does not belong to the domain - sage: G = PermutationGroup([ [('c','d')], [('a','c')] ], domain='abcd') sage: G.stabilizer('a') Subgroup generated by [('c','d')] of (Permutation Group with generators [('c','d'), ('a','c')]) sage: G.stabilizer('b') Subgroup generated by [('c','d'), ('a','c')] of (Permutation Group with generators [('c','d'), ('a','c')]) sage: G.stabilizer('c') Subgroup generated by [('a','d')] of (Permutation Group with generators [('c','d'), ('a','c')]) sage: G.stabilizer('d') Subgroup generated by [('a','c')] of (Permutation Group with generators [('c','d'), ('a','c')]) - >>> from sage.all import * >>> G = PermutationGroup([ [('c','d')], [('a','c')] ], domain='abcd') >>> G.stabilizer('a') Subgroup generated by [('c','d')] of (Permutation Group with generators [('c','d'), ('a','c')]) >>> G.stabilizer('b') Subgroup generated by [('c','d'), ('a','c')] of (Permutation Group with generators [('c','d'), ('a','c')]) >>> G.stabilizer('c') Subgroup generated by [('a','d')] of (Permutation Group with generators [('c','d'), ('a','c')]) >>> G.stabilizer('d') Subgroup generated by [('a','c')] of (Permutation Group with generators [('c','d'), ('a','c')]) 
 - strong_generating_system(base_of_group=None, implementation='sage')[source]¶
- Return a Strong Generating System of - selfaccording the given base for the right action of- selfon itself.- base_of_groupis a list of the positions on which- selfacts, in any order. The algorithm returns a list of transversals and each transversal is a list of permutations. By default,- base_of_groupis- [1, 2, 3, ..., d]where \(d\) is the degree of the group.- For - base_of_group= \([ \mathrm{pos}_1, \mathrm{pos}_2, \dots , \mathrm{pos}_d]\) let \(G_i\) be the subgroup of \(G\) =- selfwhich stabilizes \(\mathrm{pos}_1, \mathrm{pos}_2, \dots , \mathrm{pos}_i\), so\[G = G_0 \supset G_1 \supset G_2 \supset \dots \supset G_n = \{e\}\]- Then the algorithm returns \([ G_i.\mathrm{transversals}(\mathrm{pos}_{i+1})]_{1 \leq i \leq n}\) - INPUT: - base_of_group– (default:- [1, 2, 3, ..., d]) a list containing the integers \(1, 2, \ldots , d\) in any order, where \(d\) is the degree of- self
- implementation– (default:- 'sage') either- 'sage'– use the direct implementation in Sage
- 'gap'– if used, the- base_of_groupmust be- Noneand the computation is directly performed in GAP
 
 - OUTPUT: list of lists of permutations from the group, which forms a strong generating system - Warning - The outputs for implementations - 'sage'and- 'gap'differ: First, the output is reversed, and second, it might be that- 'sage'does not contain the trivial subgroup while- 'gap'does.- Also, both algorithms might yield different results based on the order in which - base_of_groupis given in the first situation.- EXAMPLES: - sage: G = PermutationGroup([[(7,8)],[(3,4)],[(4,5)]]) sage: G.strong_generating_system() [[()], [()], [(), (3,4), (3,5,4)], [(), (4,5)], [()], [()], [(), (7,8)], [()]] sage: G = PermutationGroup([[(1,2,3,4)],[(1,2)]]) sage: G.strong_generating_system() [[(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)], [(), (2,4,3), (2,3,4)], [(), (3,4)], [()]] sage: G = PermutationGroup([[(1,2,3)],[(4,5,7)],[(1,4,6)]]) sage: G.strong_generating_system() [[(), (1,2,3), (1,4,6), (1,3,2), (1,5,7,4,6), (1,6,4), (1,7,5,4,6)], [(), (2,3,6), (2,6,3), (2,7,5,6,3), (2,5,6,3)(4,7), (2,4,5,6,3)], [(), (3,5,6), (3,4,7,5,6), (3,6)(5,7), (3,7,4,5,6)], [(), (4,7,5), (4,5,7), (4,6,7)], [(), (5,6,7), (5,7,6)], [()], [()]] sage: G = PermutationGroup([[(1,2,3)],[(2,3,4)],[(3,4,5)]]) sage: G.strong_generating_system([5,4,3,2,1]) [[(), (1,5,3,4,2), (1,5,4,3,2), (1,5)(2,3), (1,5,2)], [(1,4)(2,3), (1,4,3), (1,4,2), ()], [(1,2,3), (1,3,2), ()], [()], [()]] sage: G = PermutationGroup([[(3,4)]]) sage: G.strong_generating_system() [[()], [()], [(), (3,4)], [()]] sage: G.strong_generating_system(base_of_group=[3,1,2,4]) [[(), (3,4)], [()], [()], [()]] sage: G = TransitiveGroup(12,17) sage: G.strong_generating_system() [[(), (1,4,11,2)(3,6,5,8)(7,10,9,12), (1,8,3,2)(4,11,10,9)(5,12,7,6), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,12,7,2)(3,10,9,8)(4,11,6,5), (1,11)(2,8)(3,5)(4,10)(6,12)(7,9), (1,10,11,8)(2,3,12,5)(4,9,6,7), (1,3)(2,8)(4,10)(5,7)(6,12)(9,11), (1,2,3,8)(4,9,10,11)(5,6,7,12), (1,6,7,8)(2,3,4,9)(5,10,11,12), (1,5,9)(3,11,7), (1,9,5)(3,7,11)], [(), (2,6,10)(4,12,8), (2,10,6)(4,8,12)], [()], [()], [()], [()], [()], [()], [()], [()], [()], [()]] sage: A = PermutationGroup([(1,2),(1,2,3,4,5,6,7,8,9)]) sage: X = A.strong_generating_system() sage: Y = A.strong_generating_system(implementation='gap') sage: [len(x) for x in X] [9, 8, 7, 6, 5, 4, 3, 2, 1] sage: [len(y) for y in Y] [1, 2, 3, 4, 5, 6, 7, 8, 9] - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(7),Integer(8))],[(Integer(3),Integer(4))],[(Integer(4),Integer(5))]]) >>> G.strong_generating_system() [[()], [()], [(), (3,4), (3,5,4)], [(), (4,5)], [()], [()], [(), (7,8)], [()]] >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3),Integer(4))],[(Integer(1),Integer(2))]]) >>> G.strong_generating_system() [[(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)], [(), (2,4,3), (2,3,4)], [(), (3,4)], [()]] >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3))],[(Integer(4),Integer(5),Integer(7))],[(Integer(1),Integer(4),Integer(6))]]) >>> G.strong_generating_system() [[(), (1,2,3), (1,4,6), (1,3,2), (1,5,7,4,6), (1,6,4), (1,7,5,4,6)], [(), (2,3,6), (2,6,3), (2,7,5,6,3), (2,5,6,3)(4,7), (2,4,5,6,3)], [(), (3,5,6), (3,4,7,5,6), (3,6)(5,7), (3,7,4,5,6)], [(), (4,7,5), (4,5,7), (4,6,7)], [(), (5,6,7), (5,7,6)], [()], [()]] >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3))],[(Integer(2),Integer(3),Integer(4))],[(Integer(3),Integer(4),Integer(5))]]) >>> G.strong_generating_system([Integer(5),Integer(4),Integer(3),Integer(2),Integer(1)]) [[(), (1,5,3,4,2), (1,5,4,3,2), (1,5)(2,3), (1,5,2)], [(1,4)(2,3), (1,4,3), (1,4,2), ()], [(1,2,3), (1,3,2), ()], [()], [()]] >>> G = PermutationGroup([[(Integer(3),Integer(4))]]) >>> G.strong_generating_system() [[()], [()], [(), (3,4)], [()]] >>> G.strong_generating_system(base_of_group=[Integer(3),Integer(1),Integer(2),Integer(4)]) [[(), (3,4)], [()], [()], [()]] >>> G = TransitiveGroup(Integer(12),Integer(17)) >>> G.strong_generating_system() [[(), (1,4,11,2)(3,6,5,8)(7,10,9,12), (1,8,3,2)(4,11,10,9)(5,12,7,6), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,12,7,2)(3,10,9,8)(4,11,6,5), (1,11)(2,8)(3,5)(4,10)(6,12)(7,9), (1,10,11,8)(2,3,12,5)(4,9,6,7), (1,3)(2,8)(4,10)(5,7)(6,12)(9,11), (1,2,3,8)(4,9,10,11)(5,6,7,12), (1,6,7,8)(2,3,4,9)(5,10,11,12), (1,5,9)(3,11,7), (1,9,5)(3,7,11)], [(), (2,6,10)(4,12,8), (2,10,6)(4,8,12)], [()], [()], [()], [()], [()], [()], [()], [()], [()], [()]] >>> A = PermutationGroup([(Integer(1),Integer(2)),(Integer(1),Integer(2),Integer(3),Integer(4),Integer(5),Integer(6),Integer(7),Integer(8),Integer(9))]) >>> X = A.strong_generating_system() >>> Y = A.strong_generating_system(implementation='gap') >>> [len(x) for x in X] [9, 8, 7, 6, 5, 4, 3, 2, 1] >>> [len(y) for y in Y] [1, 2, 3, 4, 5, 6, 7, 8, 9] 
 - structure_description(G, latex=False)[source]¶
- Return a string that tries to describe the structure of - G.- This methods wraps GAP’s - StructureDescriptionmethod.- For full details, including the form of the returned string and the algorithm to build it, see GAP’s documentation. - INPUT: - latex– boolean (default:- False); if- True, return a LaTeX formatted string
 - OUTPUT: string - Warning - From GAP’s documentation: The string returned by - StructureDescriptionis not an isomorphism invariant: non-isomorphic groups can have the same string value, and two isomorphic groups in different representations can produce different strings.- EXAMPLES: - sage: # needs sage.groups sage: G = CyclicPermutationGroup(6) sage: G.structure_description() 'C6' sage: G.structure_description(latex=True) 'C_{6}' sage: G2 = G.direct_product(G, maps=False) sage: LatexExpr(G2.structure_description(latex=True)) C_{6} \times C_{6} - >>> from sage.all import * >>> # needs sage.groups >>> G = CyclicPermutationGroup(Integer(6)) >>> G.structure_description() 'C6' >>> G.structure_description(latex=True) 'C_{6}' >>> G2 = G.direct_product(G, maps=False) >>> LatexExpr(G2.structure_description(latex=True)) C_{6} \times C_{6} - This method is mainly intended for small groups or groups with few normal subgroups. Even then there are some surprises: - sage: D3 = DihedralGroup(3) # needs sage.groups sage: D3.structure_description() # needs sage.groups 'S3' - >>> from sage.all import * >>> D3 = DihedralGroup(Integer(3)) # needs sage.groups >>> D3.structure_description() # needs sage.groups 'S3' - We use the Sage notation for the degree of dihedral groups: - sage: D4 = DihedralGroup(4) # needs sage.groups sage: D4.structure_description() # needs sage.groups 'D4' - >>> from sage.all import * >>> D4 = DihedralGroup(Integer(4)) # needs sage.groups >>> D4.structure_description() # needs sage.groups 'D4' - Works for finitely presented groups (Issue #17573): - sage: F.<x, y> = FreeGroup() # needs sage.groups sage: G = F / [x^2*y^-1, x^3*y^2, x*y*x^-1*y^-1] # needs sage.groups sage: G.structure_description() # needs sage.groups 'C7' - >>> from sage.all import * >>> F = FreeGroup(names=('x', 'y',)); (x, y,) = F._first_ngens(2)# needs sage.groups >>> G = F / [x**Integer(2)*y**-Integer(1), x**Integer(3)*y**Integer(2), x*y*x**-Integer(1)*y**-Integer(1)] # needs sage.groups >>> G.structure_description() # needs sage.groups 'C7' - And matrix groups (Issue #17573): - sage: groups.matrix.GL(4,2).structure_description() # needs sage.libs.gap sage.modules 'A8' - >>> from sage.all import * >>> groups.matrix.GL(Integer(4),Integer(2)).structure_description() # needs sage.libs.gap sage.modules 'A8' 
 - subgroup(gens=None, gap_group=None, domain=None, category=None, canonicalize=True, check=True)[source]¶
- Wraps the - PermutationGroup_subgroupconstructor. The argument- gensis a list of elements of- self.- EXAMPLES: - sage: G = PermutationGroup([(1,2,3),(3,4,5)]) sage: g = G((1,2,3)) sage: G.subgroup([g]) Subgroup generated by [(1,2,3)] of (Permutation Group with generators [(3,4,5), (1,2,3)]) - >>> from sage.all import * >>> G = PermutationGroup([(Integer(1),Integer(2),Integer(3)),(Integer(3),Integer(4),Integer(5))]) >>> g = G((Integer(1),Integer(2),Integer(3))) >>> G.subgroup([g]) Subgroup generated by [(1,2,3)] of (Permutation Group with generators [(3,4,5), (1,2,3)]) 
 - subgroups()[source]¶
- Return a list of all the subgroups of - self.- OUTPUT: - Each possible subgroup of - selfis contained once in the returned list. The list is in order, according to the size of the subgroups, from the trivial subgroup with one element on through up to the whole group. Conjugacy classes of subgroups are contiguous in the list.- Warning - For even relatively small groups this method can take a very long time to execute, or create vast amounts of output. Likely both. Its purpose is instructional, as it can be useful for studying small groups. The 156 subgroups of the full symmetric group on 5 symbols of order 120, \(S_5\), can be computed in about a minute on commodity hardware in 2011. The 64 subgroups of the cyclic group of order \(30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13\) takes about twice as long. - For faster results, which still exhibit the structure of the possible subgroups, use - conjugacy_classes_subgroups().- EXAMPLES: - sage: G = SymmetricGroup(3) sage: G.subgroups() [Subgroup generated by [()] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(2,3)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,2)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,3)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,2,3)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,2,3), (2,3)] of (Symmetric group of order 3! as a permutation group)] sage: G = CyclicPermutationGroup(14) sage: G.subgroups() [Subgroup generated by [()] of (Cyclic group of order 14 as a permutation group), Subgroup generated by [(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)] of (Cyclic group of order 14 as a permutation group), Subgroup generated by [(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)] of (Cyclic group of order 14 as a permutation group), Subgroup generated by [(1,3,5,7,9,11,13)(2,4,6,8,10,12,14), (1,2,3,4,5,6,7,8,9,10,11,12,13,14)] of (Cyclic group of order 14 as a permutation group)] - >>> from sage.all import * >>> G = SymmetricGroup(Integer(3)) >>> G.subgroups() [Subgroup generated by [()] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(2,3)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,2)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,3)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,2,3)] of (Symmetric group of order 3! as a permutation group), Subgroup generated by [(1,2,3), (2,3)] of (Symmetric group of order 3! as a permutation group)] >>> G = CyclicPermutationGroup(Integer(14)) >>> G.subgroups() [Subgroup generated by [()] of (Cyclic group of order 14 as a permutation group), Subgroup generated by [(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)] of (Cyclic group of order 14 as a permutation group), Subgroup generated by [(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)] of (Cyclic group of order 14 as a permutation group), Subgroup generated by [(1,3,5,7,9,11,13)(2,4,6,8,10,12,14), (1,2,3,4,5,6,7,8,9,10,11,12,13,14)] of (Cyclic group of order 14 as a permutation group)] - AUTHOR: - Rob Beezer (2011-01-24) 
 
 - sylow_subgroup(p)[source]¶
- Return a Sylow \(p\)-subgroup of the finite group \(G\), where \(p\) is a prime. - This is a \(p\)-subgroup of \(G\) whose index in \(G\) is coprime to \(p\). - Wraps the GAP function - SylowSubgroup.- EXAMPLES: - sage: G = PermutationGroup(['(1,2,3)', '(2,3)']) sage: G.sylow_subgroup(2) Subgroup generated by [(2,3)] of (Permutation Group with generators [(2,3), (1,2,3)]) sage: G.sylow_subgroup(5) Subgroup generated by [()] of (Permutation Group with generators [(2,3), (1,2,3)]) - >>> from sage.all import * >>> G = PermutationGroup(['(1,2,3)', '(2,3)']) >>> G.sylow_subgroup(Integer(2)) Subgroup generated by [(2,3)] of (Permutation Group with generators [(2,3), (1,2,3)]) >>> G.sylow_subgroup(Integer(5)) Subgroup generated by [()] of (Permutation Group with generators [(2,3), (1,2,3)]) 
 - transversals(point)[source]¶
- If G is a permutation group acting on the set \(X = \{1, 2, ...., n\}\) and H is the stabilizer subgroup of <integer>, a right (respectively left) transversal is a set containing exactly one element from each right (respectively left) coset of H. This method returns a right transversal of - selfby the stabilizer of- selfon <integer> position.- EXAMPLES: - sage: G = PermutationGroup([ [(3,4)], [(1,3)] ]) sage: G.transversals(1) [(), (1,3,4), (1,4,3)] sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]]) sage: G.transversals(1) [(), (1,2)(3,4), (1,3,2,10,4), (1,4,2,10,3), (1,10,4,3,2)] sage: G = PermutationGroup([ [('c','d')], [('a','c')] ]) sage: G.transversals('a') [(), ('a','c','d'), ('a','d','c')] - >>> from sage.all import * >>> G = PermutationGroup([ [(Integer(3),Integer(4))], [(Integer(1),Integer(3))] ]) >>> G.transversals(Integer(1)) [(), (1,3,4), (1,4,3)] >>> G = PermutationGroup([[(Integer(1),Integer(2)),(Integer(3),Integer(4))], [(Integer(1),Integer(2),Integer(3),Integer(4),Integer(10))]]) >>> G.transversals(Integer(1)) [(), (1,2)(3,4), (1,3,2,10,4), (1,4,2,10,3), (1,10,4,3,2)] >>> G = PermutationGroup([ [('c','d')], [('a','c')] ]) >>> G.transversals('a') [(), ('a','c','d'), ('a','d','c')] 
 - trivial_character()[source]¶
- Return the trivial character of - self.- EXAMPLES: - sage: SymmetricGroup(3).trivial_character() # needs sage.rings.number_field Character of Symmetric group of order 3! as a permutation group - >>> from sage.all import * >>> SymmetricGroup(Integer(3)).trivial_character() # needs sage.rings.number_field Character of Symmetric group of order 3! as a permutation group 
 - upper_central_series()[source]¶
- Return the upper central series of this group as a list of permutation groups. - EXAMPLES: - These computations use pseudo-random numbers, so we set the seed for reproducible testing: - sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: G.upper_central_series() [Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])] - >>> from sage.all import * >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]]) >>> G.upper_central_series() [Subgroup generated by [()] of (Permutation Group with generators [(3,4), (1,2,3)(4,5)])] 
 
- class sage.groups.perm_gps.permgroup.PermutationGroup_subgroup(ambient, gens=None, gap_group=None, domain=None, category=None, canonicalize=True, check=True)[source]¶
- Bases: - PermutationGroup_generic- Subgroup subclass of - PermutationGroup_generic, so instance methods are inherited.- EXAMPLES: - sage: G = CyclicPermutationGroup(4) sage: gens = G.gens() sage: H = DihedralGroup(4) sage: H.subgroup(gens) Subgroup generated by [(1,2,3,4)] of (Dihedral group of order 8 as a permutation group) sage: K = H.subgroup(gens) sage: K.list() [(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)] sage: K.ambient_group() Dihedral group of order 8 as a permutation group sage: K.gens() ((1,2,3,4),) - >>> from sage.all import * >>> G = CyclicPermutationGroup(Integer(4)) >>> gens = G.gens() >>> H = DihedralGroup(Integer(4)) >>> H.subgroup(gens) Subgroup generated by [(1,2,3,4)] of (Dihedral group of order 8 as a permutation group) >>> K = H.subgroup(gens) >>> K.list() [(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)] >>> K.ambient_group() Dihedral group of order 8 as a permutation group >>> K.gens() ((1,2,3,4),) - ambient_group()[source]¶
- Return the ambient group related to - self.- EXAMPLES: - An example involving the dihedral group on four elements, \(D_8\): - sage: G = DihedralGroup(4) sage: H = CyclicPermutationGroup(4) sage: gens = H.gens() sage: S = PermutationGroup_subgroup(G, list(gens)) sage: S.ambient_group() Dihedral group of order 8 as a permutation group sage: S.ambient_group() == G True - >>> from sage.all import * >>> G = DihedralGroup(Integer(4)) >>> H = CyclicPermutationGroup(Integer(4)) >>> gens = H.gens() >>> S = PermutationGroup_subgroup(G, list(gens)) >>> S.ambient_group() Dihedral group of order 8 as a permutation group >>> S.ambient_group() == G True 
 - is_normal(other=None)[source]¶
- Return - Trueif this group is a normal subgroup of- other. If- otheris not specified, then it is assumed to be the ambient group.- EXAMPLES: - sage: S = SymmetricGroup(['a','b','c']) sage: H = S.subgroup([('a', 'b', 'c')]); H Subgroup generated by [('a','b','c')] of (Symmetric group of order 3! as a permutation group) sage: H.is_normal() True - >>> from sage.all import * >>> S = SymmetricGroup(['a','b','c']) >>> H = S.subgroup([('a', 'b', 'c')]); H Subgroup generated by [('a','b','c')] of (Symmetric group of order 3! as a permutation group) >>> H.is_normal() True 
 
- sage.groups.perm_gps.permgroup.direct_product_permgroups(P)[source]¶
- Take the direct product of the permutation groups listed in - P.- EXAMPLES: - sage: G1 = AlternatingGroup([1,2,4,5]) sage: G2 = AlternatingGroup([3,4,6,7]) sage: D = direct_product_permgroups([G1,G2,G1]) sage: D.order() 1728 sage: D = direct_product_permgroups([G1]) sage: D == G1 True sage: direct_product_permgroups([]) Symmetric group of order 0! as a permutation group - >>> from sage.all import * >>> G1 = AlternatingGroup([Integer(1),Integer(2),Integer(4),Integer(5)]) >>> G2 = AlternatingGroup([Integer(3),Integer(4),Integer(6),Integer(7)]) >>> D = direct_product_permgroups([G1,G2,G1]) >>> D.order() 1728 >>> D = direct_product_permgroups([G1]) >>> D == G1 True >>> direct_product_permgroups([]) Symmetric group of order 0! as a permutation group 
- sage.groups.perm_gps.permgroup.from_gap_list(G, src)[source]¶
- Convert a string giving a list of GAP permutations into a list of elements of - G.- EXAMPLES: - sage: from sage.groups.perm_gps.permgroup import from_gap_list sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]]) sage: L = from_gap_list(G, "[(1,2,3)(4,5), (3,4)]"); L [(1,2,3)(4,5), (3,4)] sage: L[0].parent() is G True sage: L[1].parent() is G True - >>> from sage.all import * >>> from sage.groups.perm_gps.permgroup import from_gap_list >>> G = PermutationGroup([[(Integer(1),Integer(2),Integer(3)),(Integer(4),Integer(5))],[(Integer(3),Integer(4))]]) >>> L = from_gap_list(G, "[(1,2,3)(4,5), (3,4)]"); L [(1,2,3)(4,5), (3,4)] >>> L[Integer(0)].parent() is G True >>> L[Integer(1)].parent() is G True 
- sage.groups.perm_gps.permgroup.hap_decorator(f)[source]¶
- A decorator for permutation group methods that require HAP. It checks to see that HAP is installed as well as checks that the argument - pis either 0 or prime.- EXAMPLES: - sage: # optional - gap_package_hap sage: from sage.groups.perm_gps.permgroup import hap_decorator sage: def foo(self, n, p=0): print("Done") sage: foo = hap_decorator(foo) sage: foo(None, 3) Done sage: foo(None, 3, 0) Done sage: foo(None, 3, 5) Done sage: foo(None, 3, 4) Traceback (most recent call last): ... ValueError: p must be 0 or prime - >>> from sage.all import * >>> # optional - gap_package_hap >>> from sage.groups.perm_gps.permgroup import hap_decorator >>> def foo(self, n, p=Integer(0)): print("Done") >>> foo = hap_decorator(foo) >>> foo(None, Integer(3)) Done >>> foo(None, Integer(3), Integer(0)) Done >>> foo(None, Integer(3), Integer(5)) Done >>> foo(None, Integer(3), Integer(4)) Traceback (most recent call last): ... ValueError: p must be 0 or prime 
- sage.groups.perm_gps.permgroup.load_hap()[source]¶
- Load the GAP hap package into the default GAP interpreter interface. - EXAMPLES: - sage: sage.groups.perm_gps.permgroup.load_hap() # optional - gap_package_hap - >>> from sage.all import * >>> sage.groups.perm_gps.permgroup.load_hap() # optional - gap_package_hap