GeneralLinearGroup - Maple Help

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GroupTheory

 GeneralLinearGroup
 construct a permutation group isomorphic to the General Linear Group over a finite field

 Calling Sequence GeneralLinearGroup(n, q) GL( n, q )

Parameters

 n - a positive integer q - power of a prime number

Description

 • The general linear group $GL\left(n,q\right)$ is the set of all nonsingular $n×n$ 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  $GL\left(n,q\right)$ for the implemented ranges of the parameters n and q. Otherwise, a symbolic group is returned, for which Maple can do some limited computations.
 • The implemented ranges for n and q are as follows:

 $n=2$ $q\le 100$ $n=3$ $q\le 20$ $n=4$ $q\le 10$ $n=5$ $q\le 5$ $n=6,7,8,9,10$ $q=2$

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{GeneralLinearGroup}\left(2,3\right)$
 ${\mathbf{GL}}\left({2}{,}{3}\right)$ (1)
 > $G≔\mathrm{GL}\left(2,5\right)$
 ${G}{≔}{\mathbf{GL}}\left({2}{,}{5}\right)$ (2)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${480}$ (3)
 > $\mathrm{cs}≔\mathrm{CompositionSeries}\left(G\right)$
 ${\mathrm{cs}}{≔}{\mathbf{GL}}\left({2}{,}{5}\right){▹}{\mathrm{< a permutation group on 24 letters with 5 generators >}}{▹}{\dots }{▹}⟨\left({1}{,}{3}\right)\left({2}{,}{4}\right)\left({5}{,}{15}\right)\left({6}{,}{18}\right)\left({7}{,}{19}\right)\left({8}{,}{16}\right)\left({9}{,}{17}\right)\left({10}{,}{20}\right)\left({11}{,}{23}\right)\left({12}{,}{24}\right)\left({13}{,}{21}\right)\left({14}{,}{22}\right)⟩{▹}⟨⟩$ (4)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(S\right),S=\mathrm{cs}\right)$
 ${480}{,}{240}{,}{120}{,}{2}{,}{1}$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{GL}\left(4,3\right)\right)$
 ${24261120}$ (6)
 > $\mathrm{ClassNumber}\left(\mathrm{GL}\left(31,q\right)\right)$
 ${{q}}^{{31}}{-}{{q}}^{{15}}{-}{{q}}^{{14}}{-}{{q}}^{{13}}{-}{{q}}^{{12}}{-}{{q}}^{{11}}{-}{{q}}^{{10}}{+}{2}{}{{q}}^{{8}}{+}{3}{}{{q}}^{{7}}{+}{4}{}{{q}}^{{6}}{+}{{q}}^{{5}}{-}{3}{}{{q}}^{{4}}{-}{3}{}{{q}}^{{3}}{+}{q}$ (7)
 > $\mathrm{GroupOrder}\left(\mathrm{GL}\left(n,q\right)\right)$
 ${\prod }_{{k}{=}{0}}^{{n}{-}{1}}{}\left({{q}}^{{n}}{-}{{q}}^{{k}}\right)$ (8)
 > $\mathrm{GroupOrder}\left(\mathrm{GL}\left(3,q\right)\right)$
 $\left({{q}}^{{3}}{-}{1}\right){}\left({{q}}^{{3}}{-}{q}\right){}\left({{q}}^{{3}}{-}{{q}}^{{2}}\right)$ (9)
 > $\mathrm{GroupOrder}\left(\mathrm{DerivedSubgroup}\left(\mathrm{GL}\left(n,q\right)\right)\right)$
 ${{q}}^{\left(\genfrac{}{}{0}{}{{n}}{{2}}\right)}{}\left({\prod }_{{k}{=}{1}}^{{n}{-}{1}}{}\left({{q}}^{{k}{+}{1}}{-}{1}\right)\right)$ (10)

Compatibility

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