GroupTheory/ProjectiveGeneralSemilinearGroup - Maple Help

GroupTheory

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

 Calling Sequence ProjectiveGeneralSemilinearGroup( n, q ) PGammaL( n, q )

Parameters

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

Description

 • The projective general semi-linear group $P\Gamma L\left(n,q\right)$ is the quotient of the group $\Gamma L\left(n,q\right)$ of all semi-linear transformations of an $n$-dimensional vector space over the field with $q$ elements, by the center of $GL\left(n,q\right)$ . 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 $P\Gamma L\left(n,q\right)$ and $GL\left(n,q\right)$ are equal.
 • If n is a positive integer, and q is a prime power, then the ProjectiveGeneralSemilinearGroup( n, q ) command returns a permutation group isomorphic to the projective general semi-linear group $P\Gamma L\left(n,q\right)$ . Otherwise, a symbolic group is returned, with which Maple can do some limited computations.
 • The abbreviation PGammaL( n, q ) is available as a synonym for ProjectiveGeneralSemilinearGroup( n, q ).

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ProjectiveGeneralSemilinearGroup}\left(2,4\right)$
 ${G}{≔}⟨\left({2}{,}{3}{,}{4}\right){,}\left({1}{,}{2}{,}{5}\right){,}\left({3}{,}{4}\right)⟩$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${120}$ (2)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(5\right)\right)$
 ${\mathrm{true}}$ (3)
 > $G≔\mathrm{PGammaL}\left(2,5\right)$
 ${G}{≔}⟨\left({2}{,}{4}{,}{5}{,}{3}\right){,}\left({1}{,}{5}{,}{6}\right)\left({2}{,}{3}{,}{4}\right)⟩$ (4)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${120}$ (5)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(5\right)\right)$
 ${\mathrm{true}}$ (6)
 > $G≔\mathrm{PGammaL}\left(2,9\right)$
 ${G}{≔}⟨\left({2}{,}{7}{,}{5}{,}{6}{,}{3}{,}{4}{,}{9}{,}{8}\right){,}\left({1}{,}{3}{,}{10}\right)\left({4}{,}{5}{,}{7}\right)\left({6}{,}{8}{,}{9}\right){,}\left({4}{,}{8}\right)\left({5}{,}{9}\right)\left({6}{,}{7}\right)⟩$ (7)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${1440}$ (8)
 > $\mathrm{ct}≔\mathrm{CharacterTable}\left(G\right)$
 > $\mathrm{Display}\left(\mathrm{ct}\right)$

 C 1a 2a 2b 2c 3a 4a 4b 4c 5a 6a 8a 8b 10a |C| 1 30 36 45 80 90 90 180 144 240 180 180 144 $\mathrm{χ__1}$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $\mathrm{χ__2}$ $1$ $-1$ $-1$ $1$ $1$ $-1$ $1$ $1$ $1$ $-1$ $-1$ $1$ $-1$ $\mathrm{χ__3}$ $1$ $-1$ $1$ $1$ $1$ $-1$ $1$ $-1$ $1$ $-1$ $1$ $-1$ $1$ $\mathrm{χ__4}$ $1$ $1$ $-1$ $1$ $1$ $1$ $1$ $-1$ $1$ $1$ $-1$ $-1$ $-1$ $\mathrm{χ__5}$ $9$ $-3$ $-1$ $1$ $0$ $1$ $1$ $-1$ $-1$ $0$ $1$ $1$ $-1$ $\mathrm{χ__6}$ $9$ $-3$ $1$ $1$ $0$ $1$ $1$ $1$ $-1$ $0$ $-1$ $-1$ $1$ $\mathrm{χ__7}$ $9$ $3$ $-1$ $1$ $0$ $-1$ $1$ $1$ $-1$ $0$ $1$ $-1$ $-1$ $\mathrm{χ__8}$ $9$ $3$ $1$ $1$ $0$ $-1$ $1$ $-1$ $-1$ $0$ $-1$ $1$ $1$ $\mathrm{χ__9}$ $10$ $-2$ $0$ $2$ $1$ $-2$ $-2$ $0$ $0$ $1$ $0$ $0$ $0$ $\mathrm{χ__10}$ $10$ $2$ $0$ $2$ $1$ $2$ $-2$ $0$ $0$ $-1$ $0$ $0$ $0$ $\mathrm{χ__11}$ $16$ $0$ $-4$ $0$ $-2$ $0$ $0$ $0$ $1$ $0$ $0$ $0$ $1$ $\mathrm{χ__12}$ $16$ $0$ $4$ $0$ $-2$ $0$ $0$ $0$ $1$ $0$ $0$ $0$ $-1$ $\mathrm{χ__13}$ $20$ $0$ $0$ $-4$ $2$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$

 > $\mathrm{DrawNormalSubgroupLattice}\left(G\right)$
 > $\mathrm{GroupOrder}\left(\mathrm{PGammaL}\left(3,8\right)\right)$
 ${49448448}$ (9)
 > $\mathrm{GroupOrder}\left(\mathrm{PGammaL}\left(n,q\right)\right)$
 $\frac{{\mathrm{logp}}{}\left({q}\right){}\left({\prod }_{{k}{=}{0}}^{{n}{-}{1}}{}\left({{q}}^{{n}}{-}{{q}}^{{k}}\right)\right)}{{q}{-}{1}}$ (10)
 > $\mathrm{GroupOrder}\left(\mathrm{PGammaL}\left(5,q\right)\right)$
 $\frac{{\mathrm{logp}}{}\left({q}\right){}\left({{q}}^{{5}}{-}{1}\right){}\left({{q}}^{{5}}{-}{q}\right){}\left({{q}}^{{5}}{-}{{q}}^{{2}}\right){}\left({{q}}^{{5}}{-}{{q}}^{{3}}\right){}\left({{q}}^{{5}}{-}{{q}}^{{4}}\right)}{{q}{-}{1}}$ (11)