SpecialOrthogonalGroup - Maple Help

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GroupTheory

 SpecialOrthogonalGroup
 construct a permutation group isomorphic to a special orthogonal group

 Calling Sequence SpecialOrthogonalGroup(d, n, q)

Parameters

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

Description

 • The special orthogonal group $SO\left(d,n,q\right)$ is the set of all $n×n$ 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 $-1$ 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:

 $n=2$ $q\le 100$ $n=3$ $q\le 20$ $n=4$ $q\le 10$ $n=5$ $q\le 5$ $n=6,7,8$ $q=3$

 • 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 $-1$.)
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $\mathrm{SpecialOrthogonalGroup}\left(0,9,2\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{GeneralOrthogonalGroup}}{}\left({0}{,}{9}{,}{2}\right)$ (1)
 > $G≔\mathrm{SpecialOrthogonalGroup}\left(1,4,7\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{SpecialOrthogonalGroup}}{}\left({1}{,}{4}{,}{7}\right)$ (2)
 > $\mathrm{Degree}\left(G\right)$
 ${128}$ (3)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${112896}$ (4)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{true}}$ (5)
 > $G≔\mathrm{SpecialOrthogonalGroup}\left(-1,4,7\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{SpecialOrthogonalGroup}}{}\left({-}{1}{,}{4}{,}{7}\right)$ (6)
 > $\mathrm{Degree}\left(G\right)$
 ${100}$ (7)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${117600}$ (8)
 > $\mathrm{IsTransitive}\left(G\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{GroupOrder}\left(\mathrm{SpecialOrthogonalGroup}\left(0,7,3\right)\right)$
 ${9170703360}$ (10)
 > $\mathrm{IsSimple}\left(\mathrm{DerivedSubgroup}\left(\mathrm{SpecialOrthogonalGroup}\left(-1,4,8\right)\right)\right)$
 ${\mathrm{true}}$ (11)
 > $\mathrm{IsSimple}\left(\mathrm{DerivedSubgroup}\left(\mathrm{SpecialOrthogonalGroup}\left(1,4,8\right)\right)\right)$
 ${\mathrm{false}}$ (12)
 > $G≔\mathrm{SpecialOrthogonalGroup}\left(0,5,q\right)$
 ${\mathrm{GroupTheory}}{:-}{\mathrm{SpecialOrthogonalGroup}}{}\left({0}{,}{5}{,}{q}\right)$ (13)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${{q}}^{{4}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{-}{1}\right)$ (14)
 > $\mathrm{Display}\left(\mathrm{CharacterTable}\left(\mathrm{SpecialOrthogonalGroup}\left(1,4,3\right)\right)\right)$

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