The projective special linear group $PSL\left(n\,q\right)$ is the quotient of the special linear group $SL\left(n\,q\right)$ by its center.

The ProjectiveSpecialLinearGroup( n, q ) command returns a permutation group isomorphic to the projective special linear group $PSL\left(n\,q\right)$ .

If either, or both, of n and q is non-numeric, then a symbolic group representing the symplectic group is returned.

The command PSL( n, q ) is provided as an abbreviation.

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

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

$\mathrm{ProjectiveSpecialLinearGroup}\left(3\,2\right)$

$\mathbf{PSL}\left(3\,2\right)$

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

$5616$

$G≔\mathrm{PSL}\left(3\,4\right)\:$

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

$20160$

$\mathrm{GroupOrder}\left(\mathrm{Alt}\left(8\right)\right)$

$20160$

$p≔\mathrm{Perm}\left(\left[\left[1\,2\,3\,4\,5\right]\,\left[6\,7\,8\right]\right]\right)$

$p≔\left(1\,2\,3\,4\,5\right)\left(6\,7\,8\right)$

$\mathrm{PermOrder}\left(p\right)$

$15$

$\mathrm{ormap}\left(g\mapsto \mathrm{PermOrder}\left(g\right)=15\,\mathrm{Elements}\left(G\right)\right)$

$\mathrm{false}$

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

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{Alt}\left(8\right)\right)$

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{PSL}\left(2\,3\right)\,\mathrm{Alt}\left(4\right)\right)$

$\mathrm{true}$

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

$\mathrm{true}$

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

$\mathrm{true}$

$\mathrm{AreIsomorphic}\left(\mathrm{PSL}\left(2\,9\right)\,\mathrm{Alt}\left(6\right)\right)$

$\mathrm{true}$

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

$\frac{q^6\left(q^2-1\right)\left(q^3-1\right)\left(q^4-1\right)}{\mathrm{igcd}\left(4\,q-1\right)}$

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

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

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