EXAMPLES:
sage: GL(4,QQ)
General Linear Group of degree 4 over Rational Field
sage: GL(1,ZZ)
General Linear Group of degree 1 over Integer Ring
sage: GL(100,RR)
General Linear Group of degree 100 over Real Field with 53 bits of precision
sage: GL(3,GF(49,'a'))
General Linear Group of degree 3 over Finite Field in a of size 7^2
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 2] [0 1] [2 0] [0 2] [0 1] [0 2]
[0 1], [1 1], [2 1], [0 2], [1 2], [2 2], [1 0]
)
sage: G = SL(6,GF(5))
sage: G.gens()
(
[2 0 0 0 0 0] [4 0 0 0 0 1]
[0 3 0 0 0 0] [4 0 0 0 0 0]
[0 0 1 0 0 0] [0 4 0 0 0 0]
[0 0 0 1 0 0] [0 0 4 0 0 0]
[0 0 0 0 1 0] [0 0 0 4 0 0]
[0 0 0 0 0 1], [0 0 0 0 4 0]
)
AUTHORS:
REFERENCES:
Return the general linear group.
The general linear group consists of all
matrices that are invertible over the ring
.
Note
This group is also available via groups.matrix.GL().
INPUT:
EXAMPLES:
sage: G = GL(6,GF(5))
sage: G.order()
11064475422000000000000000
sage: G.base_ring()
Finite Field of size 5
sage: G.category()
Category of finite groups
sage: TestSuite(G).run()
sage: G = GL(6, QQ)
sage: G.category()
Category of groups
sage: TestSuite(G).run()
Here is the Cayley graph of (relatively small) finite General Linear Group:
sage: g = GL(2,3)
sage: d = g.cayley_graph(); d
Digraph on 48 vertices
sage: d.show(color_by_label=True, vertex_size=0.03, vertex_labels=False)
sage: d.show3d(color_by_label=True)
sage: F = GF(3); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[2,0],[0,1]]), MS([[2,1],[2,0]])]
sage: G = MatrixGroup(gens)
sage: G.order()
48
sage: G.cardinality()
48
sage: H = GL(2,F)
sage: H.order()
48
sage: H == G
True
sage: H.gens() == G.gens()
True
sage: H.as_matrix_group() == H
True
sage: H.gens()
(
[2 0] [2 1]
[0 1], [2 0]
)
TESTS:
sage: groups.matrix.GL(2, 3)
General Linear Group of degree 2 over Finite Field of size 3
Bases: sage.groups.matrix_gps.named_group.NamedMatrixGroup_gap, sage.groups.matrix_gps.linear.LinearMatrixGroup_generic
Base class for “named” matrix groups using LibGAP
INPUT:
EXAMPLES:
sage: G = GL(2, GF(3))
sage: from sage.groups.matrix_gps.named_group import NamedMatrixGroup_gap
sage: isinstance(G, NamedMatrixGroup_gap)
True
Bases: sage.groups.matrix_gps.named_group.NamedMatrixGroup_generic
Base class for “named” matrix groups
INPUT:
EXAMPLES:
sage: G = GL(2, QQ)
sage: from sage.groups.matrix_gps.named_group import NamedMatrixGroup_generic
sage: isinstance(G, NamedMatrixGroup_generic)
True
Return the special linear group.
The special linear group
consists of all
matrices that are invertible over the ring
with determinant one.
Note
This group is also available via groups.matrix.SL().
INPUT:
- n – a positive integer.
- R – ring or an integer. If an integer is specified, the corresponding finite field is used.
- var – variable used to represent generator of the finite field, if needed.
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.category() Category of finite groups 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 1] [-1 0], [0 1] )Next we compute generators for
sage: G = SL(3,ZZ); G Special Linear Group of degree 3 over Integer Ring sage: G.gens() ( [0 1 0] [ 0 1 0] [1 1 0] [0 0 1] [-1 0 0] [0 1 0] [1 0 0], [ 0 0 1], [0 0 1] ) sage: TestSuite(G).run()TESTS:
sage: groups.matrix.SL(2, 3) Special Linear Group of degree 2 over Finite Field of size 3