Special Linear Groups

AUTHORS:

  • William Stein: initial version
  • David Joyner (2006-05): added examples, _latex_, __str__, gens, as_matrix_group
  • William Stein (2006-12-09): rewrite

EXAMPLES:

sage: SL(2, ZZ)
Special Linear Group of degree 2 over Integer Ring
sage: G = SL(2,GF(3)); G
Special Linear Group of degree 2 over Finite Field of size 3
sage: G.is_finite()
True
sage: G.conjugacy_class_representatives()
[
[1 0]
[0 1],
[0 2]
[1 1],
[0 1]
[2 1],
[2 0]
[0 2],
[0 2]
[1 2],
[0 1]
[2 2],
[0 2]
[1 0]
]
sage: G = SL(6,GF(5))
sage: G.gens()
[
[2 0 0 0 0 0]
[0 3 0 0 0 0]
[0 0 1 0 0 0]
[0 0 0 1 0 0]
[0 0 0 0 1 0]
[0 0 0 0 0 1],
[4 0 0 0 0 1]
[4 0 0 0 0 0]
[0 4 0 0 0 0]
[0 0 4 0 0 0]
[0 0 0 4 0 0]
[0 0 0 0 4 0]
]
sage.groups.matrix_gps.special_linear.SL(n, R, var='a')

Return the special linear group of degree n over the ring R.

EXAMPLES:

sage: SL(3,GF(2))
Special Linear Group of degree 3 over Finite Field of size 2
sage: G = SL(15,GF(7)); G
Special Linear Group of degree 15 over Finite Field of size 7
sage: G.order()
1956712595698146962015219062429586341124018007182049478916067369638713066737882363393519966343657677430907011270206265834819092046250232049187967718149558134226774650845658791865745408000000
sage: len(G.gens())
2
sage: G = SL(2,ZZ); G
Special Linear Group of degree 2 over Integer Ring
sage: G.gens()
[
[ 0  1]
[-1  0],
[1 1]
[0 1]
]

Next we compute generators for \mathrm{SL}_3(\ZZ).

sage: G = SL(3,ZZ); G
Special Linear Group of degree 3 over Integer Ring
sage: G.gens()
[
[0 1 0]
[0 0 1]
[1 0 0],
[ 0  1  0]
[-1  0  0]
[ 0  0  1],
[1 1 0]
[0 1 0]
[0 0 1]
]
class sage.groups.matrix_gps.special_linear.SpecialLinearGroup_finite_field(n, R, var='a')
class sage.groups.matrix_gps.special_linear.SpecialLinearGroup_generic(n, R, var='a')
__call__(x)

Construct a new element in this group, i.e. try to coerce x into self if at all possible.

EXAMPLES:

sage: G = SL(3, ZZ)
sage: x = [[1,0,1], [0,1,0], [0,0,1]]
sage: G(x)
[1 0 1]
[0 1 0]
[0 0 1]
__contains__(x)

Return True if x is an element of self, False otherwise.

EXAMPLES:

sage: G = SL(2, GF(101))
sage: x = [[1,1], [0,1]]
sage: x in G
True
sage: G = SL(3, ZZ)
sage: x = [[1,0,1], [0,-1,0], [0,0,1]]
sage: x in G
False
_gap_init_()

String to create this group in GAP.

EXAMPLES:

sage: G = SL(6,GF(5)); G
Special Linear Group of degree 6 over Finite Field of size 5
sage: G._gap_init_()
'SL(6, GF(5))'
_latex_()

EXAMPLES:

sage: G = SL(6,GF(5))
sage: latex(G)
\text{SL}_{6}(\Bold{F}_{5})
_repr_()

Text representation of self.

EXAMPLES:

sage: SL(6,GF(5))
Special Linear Group of degree 6 over Finite Field of size 5

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Orthogonal Linear Groups

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