22.16 Special Linear Groups

Module: sage.groups.matrix_gps.special_linear

Special Linear Groups

Author Log:

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]
]

Module-level Functions

SL( n, R, [var=a])

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

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()
195671259569814696201521906242958634112401800718204947891606736963871306673
788236339351996634365767743090701127020626583481909204625023204918796771814
9558134226774650845658791865745408000000
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 $ \SL _3(\mathbf{Z})$ .

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: SpecialLinearGroup_finite_field

class SpecialLinearGroup_finite_field

Class: SpecialLinearGroup_generic

class SpecialLinearGroup_generic

Special Functions: _gap_init_,$ \,$ _latex_,$ \,$ _repr_

_gap_init_( self)

String to create this grop in GAP.

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_( self)

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

_repr_( self)

Text representation of self.

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

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