The general semi-linear group $\Gamma L\left(n\,q\right)$ is the group of all semi-linear transformations of an $n$-dimensional vector space over the field with $q$ elements. 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 $\Gamma L\left(n\,q\right)$ and $GL\left(n\,q\right)$ are isomorphic.

If n and q are positive integers, then the GeneralSemilinearGroup( n, q ) command returns a permutation group isomorphic to the general semi-linear group  $\Gamma L\left(n\,q\right)$ . Otherwise, a symbolic group is returned, for which Maple can do some limited computations.

The abbreviation GammaL( n, q ) is available as a synonym for GeneralSemilinearGroup( n, q ).

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

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

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

$G≔\mathbf{\Gamma L}\left(2\,4\right)$

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

$360$

$G≔\mathrm{GammaL}\left(2\,5\right)$

$G≔\mathbf{\Gamma L}\left(2\,5\right)$

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

$480$

$\mathrm{AreIsomorphic}\left(G\,\mathrm{GL}\left(2\,5\right)\right)$

$\mathrm{true}$

$G≔\mathrm{GammaL}\left(2\,9\right)$

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

$\mathrm{cs}≔\mathrm{CompositionSeries}\left(G\right)$

$\mathrm{cs}≔\mathbf{\Gamma L}\left(2\,9\right)▹⟨\left(1\,73\,12\,78\,2\,38\,24\,39\right)\left(3\,48\,31\,53\,6\,69\,62\,67\right)\left(4\,13\,43\,14\,8\,26\,77\,25\right)\left(5\,59\,46\,56\,7\,34\,65\,28\right)\left(9\,30\,61\,21\,18\,60\,32\,15\right)\left(10\,22\,64\,63\,20\,17\,47\,45\right)\left(11\,68\,76\,33\,19\,52\,44\,57\right)\left(16\,55\,42\,49\,23\,29\,75\,71\right)\left(27\,40\,74\,58\,54\,80\,37\,35\right)\left(36\,70\,51\,79\,72\,50\,66\,41\right)\,\left(1\,60\,49\,59\,2\,30\,71\,34\right)\left(3\,80\,16\,73\,6\,40\,23\,38\right)\left(4\,50\,29\,48\,8\,70\,55\,69\right)\left(5\,20\,75\,26\,7\,10\,42\,13\right)\left(9\,63\,72\,27\,18\,45\,36\,54\right)\left(11\,12\,35\,61\,19\,24\,58\,32\right)\left(14\,56\,39\,53\,25\,28\,78\,67\right)\left(15\,22\,68\,65\,21\,17\,52\,46\right)\left(31\,41\,74\,57\,62\,79\,37\,33\right)\left(43\,64\,51\,76\,77\,47\,66\,44\right)\,\left(9\,36\,18\,72\right)\left(10\,37\,19\,73\right)\left(11\,38\,20\,74\right)\left(12\,39\,21\,75\right)\left(13\,40\,22\,76\right)\left(14\,41\,23\,77\right)\left(15\,42\,24\,78\right)\left(16\,43\,25\,79\right)\left(17\,44\,26\,80\right)\left(27\,63\,54\,45\right)\left(28\,64\,55\,46\right)\left(29\,65\,56\,47\right)\left(30\,66\,57\,48\right)\left(31\,67\,58\,49\right)\left(32\,68\,59\,50\right)\left(33\,69\,60\,51\right)\left(34\,70\,61\,52\right)\left(35\,71\,62\,53\right)\,\left(9\,63\,72\,27\,18\,45\,36\,54\right)\left(10\,64\,73\,28\,19\,46\,37\,55\right)\left(11\,65\,74\,29\,20\,47\,38\,56\right)\left(12\,66\,75\,30\,21\,48\,39\,57\right)\left(13\,67\,76\,31\,22\,49\,40\,58\right)\left(14\,68\,77\,32\,23\,50\,41\,59\right)\left(15\,69\,78\,33\,24\,51\,42\,60\right)\left(16\,70\,79\,34\,25\,52\,43\,61\right)\left(17\,71\,80\,35\,26\,53\,44\,62\right)⟩▹\dots ▹⟨\left(1\,2\right)\left(3\,6\right)\left(4\,8\right)\left(5\,7\right)\left(9\,18\right)\left(10\,20\right)\left(11\,19\right)\left(12\,24\right)\left(13\,26\right)\left(14\,25\right)\left(15\,21\right)\left(16\,23\right)\left(17\,22\right)\left(27\,54\right)\left(28\,56\right)\left(29\,55\right)\left(30\,60\right)\left(31\,62\right)\left(32\,61\right)\left(33\,57\right)\left(34\,59\right)\left(35\,58\right)\left(36\,72\right)\left(37\,74\right)\left(38\,73\right)\left(39\,78\right)\left(40\,80\right)\left(41\,79\right)\left(42\,75\right)\left(43\,77\right)\left(44\,76\right)\left(45\,63\right)\left(46\,65\right)\left(47\,64\right)\left(48\,69\right)\left(49\,71\right)\left(50\,70\right)\left(51\,66\right)\left(52\,68\right)\left(53\,67\right)⟩▹⟨⟩$

$\mathrm{seq}\left(\mathrm{GroupOrder}\left(S\right)\,S=\mathrm{cs}\right)$

$11520,5760,2880,1440,720,2,1$

$\mathrm{GroupOrder}\left(\mathrm{GammaL}\left(4\,4\right)\right)$

$5922201600$

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

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

$\mathrm{GroupOrder}\left(\mathrm{GammaL}\left(3\,q\right)\right)$

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