ProjectiveSpecialUnitaryGroup - Maple Help

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GroupTheory

 ProjectiveSpecialUnitaryGroup
 construct a permutation group isomorphic to a projective special unitary group

 Calling Sequence ProjectiveSpecialUnitaryGroup( n, q ) PSU( n, q )

Parameters

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

Description

 • 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 $PSU\left(n,q\right)$ for values of the parameters n and q in the implemented ranges.
 • The implemented ranges for n and q are as follows:

 $n=2$ $q\le 241$ $n=3$ $q\le 16$ $n=4$ $q\le 5$ $n=5$ $q\le 4$ $n=6$ $q\le 3$ $n=7,8$ $q=2$

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ProjectiveSpecialUnitaryGroup}\left(3,3\right)$
 ${G}{≔}{\mathbf{PSU}}\left({3}{,}{3}\right)$ (1)
 > $\mathrm{Degree}\left(G\right)$
 ${28}$ (2)
 > $\mathrm{Generators}\left(G\right)$
 $\left[\left({3}{,}{4}{,}{6}{,}{10}{,}{12}{,}{18}{,}{19}{,}{23}\right)\left({5}{,}{8}{,}{13}{,}{20}{,}{17}{,}{11}{,}{16}{,}{7}\right)\left({9}{,}{14}{,}{21}{,}{15}{,}{22}{,}{24}{,}{26}{,}{28}\right)\left({25}{,}{27}\right){,}\left({1}{,}{2}{,}{3}{,}{5}{,}{9}{,}{15}{,}{16}{,}{18}\right)\left({4}{,}{7}{,}{12}{,}{19}{,}{24}{,}{27}{,}{26}{,}{23}\right)\left({6}{,}{11}{,}{17}{,}{13}{,}{8}{,}{10}{,}{14}{,}{21}\right)\left({20}{,}{25}\right)\right]$ (3)
 > $\mathrm{IsSoluble}\left(\mathrm{PSU}\left(2,2\right)\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{AreIsomorphic}\left(\mathrm{PSU}\left(2,3\right),\mathrm{Alt}\left(4\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{IdentifyFrobeniusGroup}\left(\mathrm{PSU}\left(3,2\right)\right)$
 ${72}{,}{2}$ (6)
 > $\mathrm{GroupOrder}\left(\mathrm{PSU}\left(5,3\right)\right)$
 ${258190571520}$ (7)
 > $\mathrm{GroupOrder}\left(\mathrm{PSU}\left(4,q\right)\right)$
 $\frac{{{q}}^{{6}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{3}}{+}{1}\right){}\left({{q}}^{{4}}{-}{1}\right)}{{\mathrm{igcd}}{}\left({4}{,}{q}{+}{1}\right)}$ (8)
 > $\mathrm{IsSimple}\left(\mathrm{PSU}\left(5,q\right)\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{IsSimple}\left(\mathrm{PSU}\left(3,q\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}3
 ${\mathrm{true}}$ (10)

Compatibility

 • 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.