GroupTheory/GeneralSemilinearGroup - Maple Help

GroupTheory

 GeneralSemilinearGroup
 construct a permutation group isomorphic to the General Semi-linear Group over a finite field

 Calling Sequence GeneralSemilinearGroup( n, q ) GammaL( n, q )

Parameters

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

Description

 • The general semi-linear group $\Gamma L\left(n,q\right)$ is the group of all semi-linear transformations of an $n$-dimensional vector space over the field with $q$ elements. It is isomorphic to a semi-direct product of the general linear group $GL\left(n,q\right)$ with the Galois group of the field with $q$ elements over its prime sub-field. Thus, if $q$ is prime, then $\Gamma L\left(n,q\right)$ and $GL\left(n,q\right)$ are isomorphic.
 • If n and q are positive integers, then the GeneralSemilinearGroup( n, q ) command returns a permutation group isomorphic to the general semi-linear group  $\Gamma L\left(n,q\right)$ . Otherwise, a symbolic group is returned, for which Maple can do some limited computations.
 • The abbreviation GammaL( n, q ) is available as a synonym for GeneralSemilinearGroup( n, q ).
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{GeneralSemilinearGroup}\left(2,4\right)$
 ${G}{≔}{\mathbf{\Gamma L}}\left({2}{,}{4}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${360}$ (2)
 > $G≔\mathrm{GammaL}\left(2,5\right)$
 ${G}{≔}{\mathbf{\Gamma L}}\left({2}{,}{5}\right)$ (3)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${480}$ (4)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{GL}\left(2,5\right)\right)$
 ${\mathrm{true}}$ (5)
 > $G≔\mathrm{GammaL}\left(2,9\right)$
 ${G}{≔}{\mathbf{\Gamma L}}\left({2}{,}{9}\right)$ (6)
 > $\mathrm{cs}≔\mathrm{CompositionSeries}\left(G\right)$
 ${\mathrm{cs}}{≔}{\mathbf{\Gamma L}}\left({2}{,}{9}\right){▹}{\mathrm{< a permutation group on 80 letters with 4 generators >}}{▹}{\dots }{▹}⟨\left({1}{,}{2}\right)\left({3}{,}{6}\right)\left({4}{,}{8}\right)\left({5}{,}{7}\right)\left({9}{,}{18}\right)\left({10}{,}{20}\right)\left({11}{,}{19}\right)\left({12}{,}{24}\right)\left({13}{,}{26}\right)\left({14}{,}{25}\right)\left({15}{,}{21}\right)\left({16}{,}{23}\right)\left({17}{,}{22}\right)\left({27}{,}{54}\right)\left({28}{,}{56}\right)\left({29}{,}{55}\right)\left({30}{,}{60}\right)\left({31}{,}{62}\right)\left({32}{,}{61}\right)\left({33}{,}{57}\right)\left({34}{,}{59}\right)\left({35}{,}{58}\right)\left({36}{,}{72}\right)\left({37}{,}{74}\right)\left({38}{,}{73}\right)\left({39}{,}{78}\right)\left({40}{,}{80}\right)\left({41}{,}{79}\right)\left({42}{,}{75}\right)\left({43}{,}{77}\right)\left({44}{,}{76}\right)\left({45}{,}{63}\right)\left({46}{,}{65}\right)\left({47}{,}{64}\right)\left({48}{,}{69}\right)\left({49}{,}{71}\right)\left({50}{,}{70}\right)\left({51}{,}{66}\right)\left({52}{,}{68}\right)\left({53}{,}{67}\right)⟩{▹}⟨⟩$ (7)
 > $\mathrm{seq}\left(\mathrm{GroupOrder}\left(S\right),S=\mathrm{cs}\right)$
 ${11520}{,}{5760}{,}{2880}{,}{1440}{,}{720}{,}{2}{,}{1}$ (8)
 > $\mathrm{GroupOrder}\left(\mathrm{GammaL}\left(4,4\right)\right)$
 ${5922201600}$ (9)
 > $\mathrm{GroupOrder}\left(\mathrm{GammaL}\left(n,q\right)\right)$
 ${\mathrm{logp}}{}\left({q}\right){}\left({\prod }_{{k}{=}{0}}^{{n}{-}{1}}{}\left({{q}}^{{n}}{-}{{q}}^{{k}}\right)\right)$ (10)
 > $\mathrm{GroupOrder}\left(\mathrm{GammaL}\left(3,q\right)\right)$
 ${\mathrm{logp}}{}\left({q}\right){}\left({{q}}^{{3}}{-}{1}\right){}\left({{q}}^{{3}}{-}{q}\right){}\left({{q}}^{{3}}{-}{{q}}^{{2}}\right)$ (11)