The projective special orthogonal group $PSO\left(d\,n\,q\right)$ is the quotient of the special orthogonal group $SO\left(d\,n\,q\right)$ by its center. The value of $d$ must be $0$ for odd $n$, or $1$ or $\mathrm{-1}$ for even $n$.

The ProjectiveSpecialOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the projective special orthogonal group $PSO\left(d\,n\,q\right)$ .

The PSO( 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 $PSO\left(d\,n\,q\right)$ is returned.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{ProjectiveSpecialOrthogonalGroup}\left(-1\,2\,7\right)$

$G≔\mathbf{PSO}\left(-1\,2\,7\right)$

$\mathrm{GroupOrder}\left(G\right)$

$4$

$\mathrm{IsCyclic}\left(G\right)$

$\mathrm{true}$

$G≔\mathrm{ProjectiveSpecialOrthogonalGroup}\left(1\,2\,8\right)$

$G≔\mathbf{PSO}\left(1\,2\,8\right)$

$\mathrm{AreIsomorphic}\left(G\,\mathrm{DihedralGroup}\left(7\right)\right)$

$\mathrm{true}$

$G≔\mathrm{PSO}\left(0\,3\,3\right)$

$G≔\mathbf{PSO}\left(0\,3\,3\right)$

$\mathrm{AreIsomorphic}\left(G\,\mathrm{Symm}\left(4\right)\right)$

$\mathrm{true}$

$G≔\mathrm{PSO}\left(-1\,4\,9\right)$

$G≔\mathbf{PSO}\left(-1\,4\,9\right)$

$\mathrm{GroupOrder}\left(G\right)$

$265680$

$\mathrm{IsSimple}\left(G\right)$

$\mathrm{true}$

$G≔\mathrm{PSO}\left(1\,4\,9\right)$

$G≔\mathbf{PSO}\left(1\,4\,9\right)$

$\mathrm{GroupOrder}\left(G\right)$

$259200$

$\mathrm{IsSimple}\left(G\right)$

$\mathrm{false}$