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GroupTheory

 ProjectiveGeneralUnitaryGroup
 construct a permutation group isomorphic to a projective general unitary group

 Calling Sequence ProjectiveGeneralUnitaryGroup(n, q) PGU(n, q)

Parameters

 n - a positive integer greater than 1 q - power of a prime number

Description

 • The projective general unitary group PGU(n, q) is the quotient of the general unitary group GU(n, q) by its center.
 • The ProjectiveGeneralUnitaryGroup( n, q ) command returns a permutation group isomorphic to the projective general unitary group of degree $n$ over the field with ${q}^{2}$ elements. In general, this is not a transitive representation.
 • Note that for $n=2$ the groups PGU(n, q) and PGL(n, q) are isomorphic, so the latter is returned in this case.
 • The ranges for n and q which are implemented are as follows:

 n = 2 q <= 100 n = 3 q <= 5 n = 4 q <= 4 n = 5,6 q = 2

 • The command PGU( n, q ) is provided as an abbreviation.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{ProjectiveGeneralUnitaryGroup}\left(2,13\right)$
 ${\mathbf{PGU}}\left({2}{,}{13}\right)$ (1)
 > $G≔\mathrm{PGU}\left(4,4\right)$
 ${G}{≔}{\mathbf{PGU}}\left({4}{,}{4}\right)$ (2)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${1018368000}$ (3)
 > $\mathrm{IsTransitive}\left(\mathrm{PGU}\left(3,3\right)\right)$
 ${\mathrm{false}}$ (4)
 > $\mathrm{Orbits}\left(\mathrm{PGU}\left(3,3\right)\right)$
 $\left[{{1}}^{{\mathbf{PGU}}\left({3}{,}{3}\right)}{,}{{2}}^{{\mathbf{PGU}}\left({3}{,}{3}\right)}\right]$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{PGU}\left(4,q\right)\right)$
 $\frac{{{q}}^{{6}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{3}}{+}{1}\right){}\left({{q}}^{{4}}{-}{1}\right)}{{\mathrm{igcd}}{}\left({4}{,}{q}{+}{1}\right)}$ (6)

Compatibility

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