# 21.15 General Linear Groups

Module: sage.groups.matrix_gps.general_linear

General Linear Groups

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


Author Log:

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

Module-level Functions

 GL( n, R, [var=a])
Return the general linear group of degree over the ring .

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: GeneralLinearGroup_finite_field

class GeneralLinearGroup_finite_field

Class: GeneralLinearGroup_generic

class GeneralLinearGroup_generic

Special Functions: _gap_init_, _latex_, _repr_

 _gap_init_( self)

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


 _latex_( self)

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


 _repr_( self)
String representation of this linear group.

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