The projective special unitary group $PSU\left(n\,q\right)$ , over the field with $q^2$ elements, is the quotient of the special unitary group $SU\left(n\,q\right)$ by its center.

Note that for $n=2$ the groups $PSU\left(n\,q\right)$ and $PSL\left(n\,q\right)$ are isomorphic. These groups are soluble being isomorphic, respectively, to the symmetric group of order $6$, and the alternating group of order $12$. Furthermore, the group $PSU\left(3\,2\right)$ is a Frobenius group of order $72$ and is soluble. For all other values of $n$ and $q$, the group $PSU\left(n\,q\right)$ is simple.

The ProjectiveSpecialUnitaryGroup( n, q ) command returns a permutation group isomorphic to the projective special unitary group $PSU\left(n\,q\right)$ .

If either or both of the arguments n and q are non-numeric, then a symbolic group representing the projective special unitary group is returned.

The command PSU( 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.

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

$G≔\mathrm{ProjectiveSpecialUnitaryGroup}\left(3\,3\right)$

$G≔\mathbf{PSU}\left(3\,3\right)$

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

$28$

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

$\left[\left(3\,4\,6\,10\,12\,18\,19\,23\right)\left(5\,8\,13\,20\,17\,11\,16\,7\right)\left(9\,14\,21\,15\,22\,24\,26\,28\right)\left(25\,27\right)\,\left(1\,2\,3\,5\,9\,15\,16\,18\right)\left(4\,7\,12\,19\,24\,27\,26\,23\right)\left(6\,11\,17\,13\,8\,10\,14\,21\right)\left(20\,25\right)\right]$

$\mathrm{IsSoluble}\left(\mathrm{PSU}\left(2\,2\right)\right)$

$\mathrm{true}$

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

$\mathrm{true}$

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

$72,2$

$\mathrm{GroupOrder}\left(\mathrm{PSU}\left(5\,3\right)\right)$

$258190571520$

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

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

$\mathrm{IsSimple}\left(\mathrm{PSU}\left(5\,q\right)\right)$

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{PSU}\left(3\&comma;q\right)\right)\mathbf{assuming}3<q$

$\mathrm{true}$

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

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

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