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 ).

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{ProjectiveSymplecticSemilinearGroup}\left(2\,9\right)$

$G≔\mathbf{P\Sigma L}\left(2\,9\right)$

$\mathrm{GroupOrder}\left(G\right)$

$720$

$G≔\mathrm{PSigmap}\left(4\,25\right)$

$G≔\mathbf{P\Sigma p}\left(4\,25\right)$

$\mathrm{GroupOrder}\left(G\right)$

$95214600000000$

$\mathrm{IsSimple}\left(G\right)$

$\mathrm{false}$

$G≔\mathrm{PSigmap}\left(6\,q\right)$

$G≔\mathbf{P\Sigma p}\left(6\,q\right)$

$\mathrm{GroupOrder}\left(G\right)$

$\frac{\mathrm{logp}\left(q\right)q^9\left(q^2-1\right)\left(q^4-1\right)\left(q^6-1\right)}{\mathrm{igcd}\left(2\,q-1\right)}$