General Linear Groups

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

AUTHORS:

• David Joyner (2006-01)
• William Stein (2006-01)
• David Joyner (2006-05): added _latex_, __str__, examples
• William Stein (2006-12-09): rewrite

sage.groups.matrix_gps.general_linear.GL(n, R, var='a')

Return the general linear group of degree over the ring .

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([[0,1],[1,0]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.order()
48
sage: G.cardinality()
48
sage: H = GL(2,F)
sage: H.order()
48
sage: H == G           # Do we really want this equality?
False
sage: H.as_matrix_group() == G
True
sage: H.gens()
[
[2 0]
[0 1],
[2 1]
[2 0]
]

class sage.groups.matrix_gps.general_linear.GeneralLinearGroup_finite_field(n, R, var='a', category=None)

Bases: sage.groups.matrix_gps.general_linear.GeneralLinearGroup_generic, sage.groups.matrix_gps.matrix_group.MatrixGroup_gap_finite_field

class sage.groups.matrix_gps.general_linear.GeneralLinearGroup_generic(n, R, var='a', category=None)

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.groups.matrix_gps.matrix_group.MatrixGroup_gap

TESTS:

sage: G6 = GL(6, QQ)
sage: G6 == G6
True
sage: G6 != G6  # check that #8695 is fixed.
False