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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.
 • 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.

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)
 > $\mathrm{Generators}\left(G\right)$
 $\left[\left({2}{,}{17}\right)\left({3}{,}{27}\right)\left({4}{,}{7}\right)\left({5}{,}{24}\right)\left({6}{,}{13}\right)\left({8}{,}{11}\right)\left({9}{,}{22}\right)\left({10}{,}{21}\right)\left({12}{,}{14}\right)\left({16}{,}{20}\right)\left({18}{,}{19}\right)\left({25}{,}{26}\right){,}\left({1}{,}{3}{,}{8}{,}{17}{,}{18}{,}{5}\right)\left({2}{,}{25}{,}{9}{,}{28}{,}{23}{,}{7}\right)\left({4}{,}{10}{,}{22}\right)\left({11}{,}{12}{,}{26}{,}{24}{,}{27}{,}{15}\right)\left({13}{,}{19}{,}{16}{,}{21}{,}{20}{,}{14}\right)\right]$ (3)
 > $\mathrm{GroupOrder}\left(\mathrm{PSU}\left(5,3\right)\right)$
 ${258190571520}$ (4)
 > $\mathrm{GroupOrder}\left(\mathrm{PSU}\left(4,q\right)\right)$
 $\frac{{{q}}^{{6}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{3}}{+}{1}\right){}\left({{q}}^{{4}}{-}{1}\right)}{{\mathrm{igcd}}{}\left({4}{,}{q}{+}{1}\right)}$ (5)

Compatibility

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