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 ).

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

$G≔\mathrm{ProjectiveGeneralSemilinearGroup}\left(2\,4\right)$

$G≔⟨\left(2\,3\,4\right)\,\left(1\,2\,5\right)\,\left(3\,4\right)⟩$

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

$120$

$\mathrm{AreIsomorphic}\left(G\,\mathrm{Symm}\left(5\right)\right)$

$\mathrm{true}$

$G≔\mathrm{PGammaL}\left(2\,5\right)$

$G≔⟨\left(2\,4\,5\,3\right)\,\left(1\,5\,6\right)\left(2\,3\,4\right)⟩$

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

$120$

$\mathrm{AreIsomorphic}\left(G\,\mathrm{Symm}\left(5\right)\right)$

$\mathrm{true}$

$G≔\mathrm{PGammaL}\left(2\,9\right)$

$G≔⟨\left(2\,7\,5\,6\,3\,4\,9\,8\right)\,\left(1\,3\,10\right)\left(4\,5\,7\right)\left(6\,8\,9\right)\,\left(4\,8\right)\left(5\,9\right)\left(6\,7\right)⟩$

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

$1440$

$\mathrm{ct}≔\mathrm{CharacterTable}\left(G\right)$

$\mathrm{Display}\left(\mathrm{ct}\right)$

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

$\mathrm{GroupOrder}\left(\mathrm{PGammaL}\left(3\,8\right)\right)$

$49448448$

$\mathrm{GroupOrder}\left(\mathrm{PGammaL}\left(n\,q\right)\right)$

$\frac{\mathrm{logp}\left(q\right)\left(\prod _{k=0}^{n-1}\left(q^n-q^k\right)\right)}{q-1}$

$\mathrm{GroupOrder}\left(\mathrm{PGammaL}\left(5\,q\right)\right)$

$\frac{\mathrm{logp}\left(q\right)\left(q^5-1\right)\left(q^5-q\right)\left(q^5-q^2\right)\left(q^5-q^3\right)\left(q^5-q^4\right)}{q-1}$