GeneralUnitaryGroup - Maple Help

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GroupTheory

 GeneralUnitaryGroup
 construct a permutation group isomorphic to a general unitary group

 Calling Sequence GeneralUnitaryGroup(n, q)

Parameters

 n - a positive integer q - power of a prime number

Description

 • The general unitary group $GU\left(n,q\right)$ (often denoted by $U\left(n,q\right)$) is the group of all $n×n$ 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:

 $n=2$ $q\le 20$ $n=3$ $q\le 5$ $n=4$ $q\le 4$ $n=5,6$ $q=2$

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{GeneralUnitaryGroup}\left(2,2\right)$
 ${\mathbf{GU}}\left({2}{,}{2}\right)$ (1)
 > $\mathrm{GroupOrder}\left(\mathrm{GeneralUnitaryGroup}\left(2,4\right)\right)$
 ${300}$ (2)
 > $\mathrm{IdentifySmallGroup}\left(\mathrm{GeneralUnitaryGroup}\left(2,4\right)\right)$
 ${300}{,}{22}$ (3)
 > $\mathrm{GroupOrder}\left(\mathrm{GeneralUnitaryGroup}\left(4,q\right)\right)$
 $\left({q}{+}{1}\right){}{{q}}^{{6}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{3}}{+}{1}\right){}\left({{q}}^{{4}}{-}{1}\right)$ (4)
 > $\mathrm{simplify}\left(\mathrm{GroupOrder}\left(\mathrm{GeneralUnitaryGroup}\left(2,{3}^{k}\right)\right)\right)$
 ${{81}}^{{k}}{+}{{27}}^{{k}}{-}{{9}}^{{k}}{-}{{3}}^{{k}}$ (5)
 > $\mathrm{simplify}\left(\mathrm{ClassNumber}\left(\mathrm{GeneralUnitaryGroup}\left(2,{3}^{k}\right)\right)\right)$
 ${{9}}^{{k}}{+}{2}{}{{3}}^{{k}}{+}{1}$ (6)

Here is a general formula for the order of the general unitary group of dimension $n$ over a field of order $q$.

 > $\mathrm{GroupOrder}\left(\mathrm{GeneralUnitaryGroup}\left(n,q\right)\right)$
 $\left({q}{+}{1}\right){}{{q}}^{\frac{{n}{}\left({n}{-}{1}\right)}{{2}}}{}\left({\prod }_{{k}{=}{1}}^{{n}{-}{1}}{}\left({{q}}^{{k}{+}{1}}{-}{\left({-1}\right)}^{{k}{+}{1}}\right)\right)$ (7)

Compatibility

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