The general unitary group $GU\left(n\,q\right)$ (often denoted by $U\left(n\,q\right)$) is the group of all $n\times n$ matrices over the field with $q^2$ elements, where $q$ is a prime power, that respect a fixed nondegenerate sesquilinear form.

The GeneralUnitaryGroup( n, q ) command returns a permutation group isomorphic to the general unitary group $GU\left(n\,q\right)$ for the implemented ranges of the parameters n and q.

The implemented ranges for n and q are as follows:

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

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

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

$\mathrm{GeneralUnitaryGroup}\left(2\,2\right)$

$\mathbf{GU}\left(2\,2\right)$

$\mathrm{GroupOrder}\left(\mathrm{GeneralUnitaryGroup}\left(2\,4\right)\right)$

$300$

$\mathrm{IdentifySmallGroup}\left(\mathrm{GeneralUnitaryGroup}\left(2\,4\right)\right)$

$300,22$

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

$\left(q+1\right)q^6\left(q^2-1\right)\left(q^3+1\right)\left(q^4-1\right)$

$\mathrm{simplify}\left(\mathrm{GroupOrder}\left(\mathrm{GeneralUnitaryGroup}\left(2\,3^k\right)\right)\right)$

${81}^k+{27}^k-9^k-3^k$

$\mathrm{simplify}\left(\mathrm{ClassNumber}\left(\mathrm{GeneralUnitaryGroup}\left(2\,3^k\right)\right)\right)$

$9^k+23^k+1$

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

$\left(q+1\right)q^{\frac{n\left(n-1\right)}{2}}\left(\prod _{k=1}^{n-1}\left(q^{k+1}-{\left(\mathrm{-1}\right)}^{k+1}\right)\right)$

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

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

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