The special orthogonal group $SO\left(d\,n\,q\right)$ is the set of all $n\times n$ matrices over the field with $q$ elements that respect a non-singular quadratic form and have determinant equal to $1$. The value of $d$ must be $0$ for odd values of $n$, or $1$ or $\mathrm{-1}$ for even values of $n$. Note that for even values of $q$ the groups $SO\left(d\,n\,q\right)$ and $GO\left(d\,n\,q\right)$ are isomorphic.

The SpecialOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the special orthogonal group $SO\left(d\,n\,q\right)$ for values of the parameters d, n and q in the implemented ranges.

The implemented ranges for n and q are as follows:

If either or both of the parameters n and q is non-numeric, then a symbolic group representing the indicated special orthogonal group is returned. (The argument d must be numeric, equal to one of $0$, $1$ or $\mathrm{-1}$.)

In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

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

$\mathrm{SpecialOrthogonalGroup}\left(0\,9\,2\right)$

$\mathrm{GroupTheory}:-\mathrm{GeneralOrthogonalGroup}\left(0\,9\,2\right)$

$G≔\mathrm{SpecialOrthogonalGroup}\left(1\,4\,7\right)$

$\mathrm{GroupTheory}:-\mathrm{SpecialOrthogonalGroup}\left(1\,4\,7\right)$

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

$128$

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

$112896$

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

$\mathrm{true}$

$G≔\mathrm{SpecialOrthogonalGroup}\left(-1\,4\,7\right)$

$\mathrm{GroupTheory}:-\mathrm{SpecialOrthogonalGroup}\left(-1\,4\,7\right)$

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

$100$

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

$117600$

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

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{SpecialOrthogonalGroup}\left(0\,7\,3\right)\right)$

$9170703360$

$\mathrm{IsSimple}\left(\mathrm{DerivedSubgroup}\left(\mathrm{SpecialOrthogonalGroup}\left(-1\,4\,8\right)\right)\right)$

$\mathrm{true}$

$\mathrm{IsSimple}\left(\mathrm{DerivedSubgroup}\left(\mathrm{SpecialOrthogonalGroup}\left(1\,4\,8\right)\right)\right)$

$\mathrm{false}$

$G≔\mathrm{SpecialOrthogonalGroup}\left(0\,5\,q\right)$

$\mathrm{GroupTheory}:-\mathrm{SpecialOrthogonalGroup}\left(0\,5\,q\right)$

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

$q^4\left(q^2-1\right)\left(q^4-1\right)$

$\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{SpecialOrthogonalGroup}\left(1\,4\,3\right)\right)\right)$