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.

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

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

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

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

$30$

$G≔\mathrm{GL}\left(2\,5\right)$

$G≔\mathbf{GL}\left(2\,5\right)$

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

$480$

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

$\mathrm{cs}≔\mathbf{GL}\left(2\,5\right)▹⟨\left(1\,23\,3\,11\right)\left(2\,9\,4\,17\right)\left(5\,6\,15\,18\right)\left(7\,13\,19\,21\right)\left(8\,22\,16\,14\right)\left(10\,12\,20\,24\right)\,\left(1\,22\,3\,14\right)\left(2\,8\,4\,16\right)\left(5\,20\,15\,10\right)\left(6\,7\,18\,19\right)\left(9\,11\,17\,23\right)\left(12\,13\,24\,21\right)\,\left(5\,8\,9\,7\,6\right)\left(10\,14\,11\,13\,12\right)\left(15\,16\,17\,19\,18\right)\left(20\,22\,23\,21\,24\right)\,\left(5\,15\right)\left(6\,16\right)\left(7\,17\right)\left(8\,18\right)\left(9\,19\right)\left(10\,20\right)\left(11\,21\right)\left(12\,22\right)\left(13\,23\right)\left(14\,24\right)⟩▹\dots ▹⟨\left(1\,3\right)\left(2\,4\right)\left(5\,15\right)\left(6\,18\right)\left(7\,19\right)\left(8\,16\right)\left(9\,17\right)\left(10\,20\right)\left(11\,23\right)\left(12\,24\right)\left(13\,21\right)\left(14\,22\right)⟩▹⟨⟩$

$\mathrm{seq}\left(\mathrm{GroupOrder}\left(S\right)\,S=\mathrm{cs}\right)$

$480,240,120,2,1$

$\mathrm{GroupOrder}\left(\mathrm{GL}\left(4\,3\right)\right)$

$24261120$

$\mathrm{ClassNumber}\left(\mathrm{GL}\left(31\,q\right)\right)$

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

$\mathrm{GroupOrder}\left(\mathrm{GL}\left(n\,q\right)\right)$

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

$\mathrm{GroupOrder}\left(\mathrm{GL}\left(3\,q\right)\right)$

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

$\mathrm{GroupOrder}\left(\mathrm{DerivedSubgroup}\left(\mathrm{GL}\left(n\,q\right)\right)\right)$

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

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.