The projective general linear group $PGL\left(n\,q\right)$ is the quotient of the general linear group $GL\left(n\,q\right)$ by its center.

The ProjectiveGeneralLinearGroup( n, q ) command returns a permutation group isomorphic to the projective general linear group $PGL\left(n\,q\right)$ . The PGL(n, q) command is provided as an alias.

If either of n or q is non-numeric, then a symbolic group is returned.

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

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

$G≔\mathrm{ProjectiveGeneralLinearGroup}\left(2\,49\right)$

$G≔\mathbf{PGL}\left(2\,49\right)$

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

$\left[\left(3\,50\,49\,48\,47\,46\,45\,44\,8\,43\,42\,41\,40\,39\,38\,37\,7\,36\,35\,34\,33\,32\,31\,30\,6\,29\,28\,27\,26\,25\,24\,23\,5\,22\,21\,20\,19\,18\,17\,16\,4\,15\,14\,13\,12\,11\,10\,9\right)\,\left(1\,2\,6\right)\left(3\,4\,8\right)\left(9\,25\,33\right)\left(10\,20\,37\right)\left(11\,32\,24\right)\left(12\,15\,41\right)\left(13\,38\,16\right)\left(14\,30\,23\right)\left(17\,31\,19\right)\left(18\,44\,47\right)\left(21\,46\,43\right)\left(22\,39\,49\right)\left(26\,34\,50\right)\left(27\,48\,35\right)\left(28\,42\,40\right)\left(29\,45\,36\right)\right]$

$G≔\mathrm{ProjectiveGeneralLinearGroup}\left(5\,2\right)$

$G≔\mathbf{PGL}\left(5\,2\right)$

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

$9999360$

$\mathrm{GroupOrder}\left(\mathrm{PGL}\left(4\,3\right)\right)$

$12130560$

$\mathrm{AreIsomorphic}\left(\mathrm{PGL}\left(2\,2\right)\,\mathrm{Symm}\left(3\right)\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{PGL}\left(2\,3\right)\,\mathrm{Symm}\left(4\right)\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{PGL}\left(2\,4\right)\,\mathrm{Alt}\left(5\right)\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{PGL}\left(2\,5\right)\,\mathrm{Symm}\left(5\right)\right)$

$\mathrm{true}$

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

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

The GroupTheory[ProjectiveGeneralLinearGroup] command was introduced in Maple 17.

For more information on Maple 17 changes, see Updates in Maple 17.

The GroupTheory[ProjectiveGeneralLinearGroup] command was updated in Maple 2020.