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GroupTheory

 ProjectiveSpecialOrthogonalGroup
 construct a permutation group isomorphic to a projective special orthogonal group

 Calling Sequence ProjectiveSpecialOrthogonalGroup(d, n, q) PSO(d, n, q)

Parameters

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

Description

 • 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 $-1$ 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{ProjectiveSpecialOrthogonalGroup}\left(-1,2,7\right)$
 ${G}{≔}{\mathbf{PSO}}\left({-1}{,}{2}{,}{7}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${4}$ (2)
 > $\mathrm{IsCyclic}\left(G\right)$
 ${\mathrm{true}}$ (3)
 > $G≔\mathrm{ProjectiveSpecialOrthogonalGroup}\left(1,2,8\right)$
 ${G}{≔}{\mathbf{PSO}}\left({1}{,}{2}{,}{8}\right)$ (4)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{DihedralGroup}\left(7\right)\right)$
 ${\mathrm{true}}$ (5)
 > $G≔\mathrm{PSO}\left(0,3,3\right)$
 ${G}{≔}{\mathbf{PSO}}\left({0}{,}{3}{,}{3}\right)$ (6)
 > $\mathrm{AreIsomorphic}\left(G,\mathrm{Symm}\left(4\right)\right)$
 ${\mathrm{true}}$ (7)
 > $G≔\mathrm{PSO}\left(-1,4,9\right)$
 ${G}{≔}{\mathbf{PSO}}\left({-1}{,}{4}{,}{9}\right)$ (8)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${265680}$ (9)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{true}}$ (10)
 > $G≔\mathrm{PSO}\left(1,4,9\right)$
 ${G}{≔}{\mathbf{PSO}}\left({1}{,}{4}{,}{9}\right)$ (11)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${259200}$ (12)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{false}}$ (13)