The general linear group $GL\left(n\,q\right)$ is the set of all nonsingular $n\times n$ matrices over a finite field of size $q$, where $q$ is a prime power.

If n and q are positive integers, then the GeneralLinearGroup( n, q ) command returns a permutation group isomorphic to the general linear group  $GL\left(n\,q\right)$ . Otherwise, a symbolic group is returned, for which Maple can do some limited computations.

The abbreviation GL( n, q ) is available as a synonym for GeneralLinearGroup( n, q ).

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$\mathrm{GeneralLinearGroup}\left(2\,3\right)$

$\mathbf{GL}\left(2\,3\right)$

$\mathrm{GroupOrder}\left(\mathrm{GL}\left(1\,31\right)\right)$

$30$

$G≔\mathrm{GL}\left(2\,5\right)$

$G≔\mathbf{GL}\left(2\,5\right)$

$\mathrm{GroupOrder}\left(G\right)$

$480$

$\mathrm{cs}≔\mathrm{CompositionSeries}\left(G\right)$

$\mathrm{cs}≔\mathbf{GL}\left(2\&comma;5\right)▹\mathrm{<\; a\; permutation\; group\; on\; 24\; letters\; with\; 4\; generators\; >}▹\dots ▹⟨\left(1\&comma;3\right)\left(2\&comma;4\right)\left(5\&comma;15\right)\left(6\&comma;18\right)\left(7\&comma;19\right)\left(8\&comma;16\right)\left(9\&comma;17\right)\left(10\&comma;20\right)\left(11\&comma;23\right)\left(12\&comma;24\right)\left(13\&comma;21\right)\left(14\&comma;22\right)⟩▹⟨⟩$

$\mathrm{seq}\left(\mathrm{GroupOrder}\left(S\right)\&comma;S\&equals;\mathrm{cs}\right)$

$480,240,120,2,1$

$\mathrm{GroupOrder}\left(\mathrm{GL}\left(4\&comma;3\right)\right)$

$24261120$

$\mathrm{ClassNumber}\left(\mathrm{GL}\left(31\&comma;q\right)\right)$

$q^{31}-q^{15}-q^{14}-q^{13}-q^{12}-q^{11}-q^{10}+2q^8+3q^7+4q^6+q^5-3q^4-3q^3+q$

$\mathrm{GroupOrder}\left(\mathrm{GL}\left(n\&comma;q\right)\right)$

$\prod _{k=0}^{n-1}\left(q^n-q^k\right)$

$\mathrm{GroupOrder}\left(\mathrm{GL}\left(3\&comma;q\right)\right)$

$\left(q^3-1\right)\left(q^3-q\right)\left(q^3-q^2\right)$

$\mathrm{GroupOrder}\left(\mathrm{DerivedSubgroup}\left(\mathrm{GL}\left(n\&comma;q\right)\right)\right)$

$q^{\left(\genfrac{}{}{0ex}{}{n}{2}\right)}\left(\prod _{k=1}^{n-1}\left(q^{k+1}-1\right)\right)$

The GroupTheory[GeneralLinearGroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.

The GroupTheory[GeneralLinearGroup] command was updated in Maple 2020.