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GroupTheory

 SpecialUnitaryGroup
 construct a permutation group isomorphic to a special unitary group

 Calling Sequence SpecialUnitaryGroup(n, q)

Parameters

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

Description

 • The special unitary group $SU\left(n,q\right)$ is the set of all n x n matrices over the field with ${q}^{2}$ elements whose determinant is $1$ and respect a fixed nondegenerate sesquilinear form.
 • The SpecialUnitaryGroup( n, q ) command returns a permutation group isomorphic to the special unitary group  $SU\left(n,q\right)$ .
 • Note that for n=2 the groups $SU\left(n,q\right)$ and $SL\left(n,q\right)$ are isomorphic so the latter is returned in this case.
 • The ranges for $n$ and $q$ are as follows:

 $n=2$ $q\le 100$ $n=3$ $q\le 10$ $n=4$ $q\le 4$ $n=5$ $q\le 3$ $n=6,7$ $q=2$

 • If either or both of the parameters n and q is non-numeric, then a symbolic group representing the indicated special 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

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SpecialUnitaryGroup}\left(3,2\right)$
 ${G}{≔}{\mathbf{SU}}\left({3}{,}{2}\right)$ (1)
 > $\mathrm{Generators}\left(G\right)$
 $\left[\left({2}{,}{3}{,}{5}{,}{9}\right)\left({4}{,}{7}{,}{13}{,}{22}\right)\left({6}{,}{11}{,}{19}{,}{26}\right)\left({8}{,}{15}{,}{17}{,}{20}\right)\left({10}{,}{18}{,}{25}{,}{24}\right)\left({12}{,}{21}{,}{27}{,}{23}\right){,}\left({1}{,}{2}{,}{4}{,}{8}{,}{16}{,}{24}{,}{21}{,}{7}{,}{14}{,}{11}{,}{20}{,}{27}\right)\left({3}{,}{6}{,}{12}{,}{5}{,}{10}{,}{9}{,}{17}{,}{18}{,}{19}{,}{25}{,}{22}{,}{26}\right)\left({13}{,}{23}{,}{15}\right)\right]$ (2)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{GroupOrder}\left(\mathrm{SpecialUnitaryGroup}\left(4,2\right)\right)$
 ${25920}$ (4)
 > $G≔\mathrm{SpecialUnitaryGroup}\left(2,23\right)$
 ${G}{≔}{\mathbf{SL}}\left({2}{,}{23}\right)$ (5)
 > $\mathrm{ClassNumber}\left(G\right)$
 ${27}$ (6)
 > $\mathrm{GroupOrder}\left(\mathrm{SpecialUnitaryGroup}\left(3,q\right)\right)$
 ${{q}}^{{3}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{3}}{+}{1}\right)$ (7)
 > $\mathrm{MinPermRepDegree}\left(\mathrm{SpecialUnitaryGroup}\left(3,5\right)\right)$
 ${378}$ (8)

Compatibility

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