GroupTheory/ProjectiveSymplecticSemilinearGroup - Maple Help

GroupTheory

 ProjectiveSymplecticSemilinearGroup
 construct a permutation group isomorphic to the projective symplectic semi-linear group over a finite field

 Calling Sequence ProjectiveSymplecticSemilinearGroup(n, q) PSigmap( n, q )

Parameters

 n - : even    : an even positive integer q - : primepower    : a power of a prime number

Description

 • The projective symplectic semi-linear group $P\Sigma p\left(n,q\right)$ is the quotient of the symplectic semi-linear group $\Sigma p\left(n,q\right)$ by the centre of its subgroup $Sp\left(n,q\right)$ . The dimension $n$ must be an even positive integer. The group $P\Sigma p\left(n,q\right)$ is a semi-direct product of the projective symplectic group $PSp\left(n,q\right)$ with the Galois group of the field GF(q). Therefore, if $q$ is prime, $P\Sigma p\left(n,q\right)$ is isomorphic to $PSp\left(n,q\right)$ .
 • If n and q are positive integers, then the ProjectiveSymplecticSemilinearGroup( n, q ) command returns a permutation group isomorphic to the projective symplectic semi-linear group  $P\Sigma p\left(n,q\right)$ . Otherwise, a symbolic group is returned, with which Maple can do some limited computations.
 • The abbreviation PSigmap( n, q ) is available as a synonym for ProjectiveSymplecticSemilinearGroup( n, q ).

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ProjectiveSymplecticSemilinearGroup}\left(2,9\right)$
 ${G}{≔}{\mathbf{P\Sigma L}}\left({2}{,}{9}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${720}$ (2)
 > $G≔\mathrm{PSigmap}\left(4,25\right)$
 ${G}{≔}{\mathbf{P\Sigma p}}\left({4}{,}{25}\right)$ (3)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${95214600000000}$ (4)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{false}}$ (5)
 > $G≔\mathrm{PSigmap}\left(6,q\right)$
 ${G}{≔}{\mathbf{P\Sigma p}}\left({6}{,}{q}\right)$ (6)
 > $\mathrm{GroupOrder}\left(G\right)$
 $\frac{{\mathrm{logp}}{}\left({q}\right){}{{q}}^{{9}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{-}{1}\right){}\left({{q}}^{{6}}{-}{1}\right)}{{\mathrm{igcd}}{}\left({2}{,}{q}{-}{1}\right)}$ (7)