GroupTheory/ProjectiveSpecialSemilinearGroup - Maple Help

GroupTheory

 ProjectiveSpecialSemilinearGroup
 construct a permutation group isomorphic to the projective special semi-linear group over a finite field

 Calling Sequence ProjectiveSpecialSemilinearGroup(n, q) PSigmaL( n, q )

Parameters

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

Description

 • The projective special semi-linear group $P\Sigma L\left(n,q\right)$ is the quotient of the special semi-linear group $\Sigma L\left(n,q\right)$ by the center of $SL\left(n,q\right)$ .
 • For a positive integer n and a power q of a prime, the ProjectiveSpecialSemilinearGroup( n, q ) command returns a permutation group isomorphic to the projective special semi-linear group  $P\Sigma L\left(n,q\right)$ . Otherwise, a symbolic group is returned, with which Maple can do some limited computations.
 • If q is a prime, then $P\Sigma L\left(n,q\right)$ and $PSL\left(n,q\right)$ are the same.
 • The abbreviation PSigmaL( n, q ) is available as a synonym for ProjectiveSpecialSemilinearGroup( n, q ).

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ProjectiveSpecialSemilinearGroup}\left(1,27\right)$
 ${G}{≔}{\mathbf{P\Sigma L}}\left({1}{,}{27}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${3}$ (2)
 > $G≔\mathrm{PSigmaL}\left(2,7\right)$
 ${G}{≔}{\mathbf{P\Sigma L}}\left({2}{,}{7}\right)$ (3)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{PSL}\left(2,7\right)\right)$
 ${\mathrm{true}}$ (4)
 > $G≔\mathrm{PSigmaL}\left(2,9\right)$
 ${G}{≔}{\mathbf{P\Sigma L}}\left({2}{,}{9}\right)$ (5)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(6\right)\right)$
 ${\mathrm{true}}$ (6)
 > $G≔\mathrm{PSigmaL}\left(4,4\right)$
 ${G}{≔}{\mathbf{P\Sigma L}}\left({4}{,}{4}\right)$ (7)
 > $\mathrm{IsPrimitive}\left(G\right)$
 ${\mathrm{true}}$ (8)
 > $\mathrm{Transitivity}\left(G\right)$
 ${2}$ (9)