GroupTheory/ProjectiveGeneralOrthogonalGroup - Maple Help

GroupTheory

 ProjectiveGeneralOrthogonalGroup
 construct a permutation group isomorphic to a projective general orthogonal group

 Calling Sequence ProjectiveGeneralOrthogonalGroup(d, n, q) PGO(d, n, q)

Parameters

 d - 0, 1 or -1 n - a positive integer q - power of a prime number

Description

 • The projective general orthogonal group $PGO\left(d,n,q\right)$ is the quotient of the general orthogonal group $GO\left(d,n,q\right)$ by its center. The value of $d$ must be $0$ for odd $n$, or $1$ or $-1$ for even $n$.
 • The ProjectiveGeneralOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the general orthogonal group $GO\left(d,n,q\right)$ .
 • The PGO( d, n, q ) command is provided as an alias.
 • If the argument q is not a prime power (and is non-numeric), then a symbolic group representing $PGO\left(d,n,q\right)$ is returned.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ProjectiveGeneralOrthogonalGroup}\left(0,3,3\right)$
 ${G}{≔}{\mathbf{PGO}}\left({0}{,}{3}{,}{3}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${24}$ (2)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{true}}$ (3)
 > $\mathrm{IsDihedral}\left(\mathrm{PGO}\left(-1,2,8\right)\right)$
 ${\mathrm{true}}$ (4)
 > $\mathrm{GroupOrder}\left(\mathrm{PGO}\left(0,5,q\right)\right)$
 $\left\{\begin{array}{cc}\frac{{\mathrm{igcd}}{}\left({2}{,}{q}{-}{1}\right){}{{q}}^{{4}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{-}{1}\right)}{{2}}& {q}{::}{\mathrm{odd}}\\ {\mathrm{igcd}}{}\left({2}{,}{q}{-}{1}\right){}{{q}}^{{4}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{-}{1}\right)& {\mathrm{otherwise}}\end{array}\right\$ (5)
 > $\mathrm{IsTrivial}\left(\mathrm{PGO}\left(0,1,{31}^{10}\right)\right)$
 ${\mathrm{true}}$ (6)