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 n over the ring R.

EXAMPLES:

sage: G = GL(6,GF(5))
sage: G.order()
11064475422000000000000000
sage: G.base_ring()
Finite Field of size 5
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: H = GL(2,F)
sage: H.order()
48
sage: H == G
True
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')
class sage.groups.matrix_gps.general_linear.GeneralLinearGroup_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: This indicates that the issue from trac #1834 is resolved:

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

EXAMPLES:

sage: G = GL(6,GF(5))
sage: G._gap_init_()
'GL(6, GF(5))'
_latex_()

EXAMPLES:

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

String representation of this linear group.

EXAMPLES:

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

Previous topic

Linear Groups

Next topic

Special Linear Groups

This Page