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GroupTheory

 GeneralOrthogonalGroup
 construct a permutation group isomorphic to a general orthogonal group

 Calling Sequence GeneralOrthogonalGroup(d, n, q)

Parameters

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

Description

 • The general orthogonal group $GO\left(d,n,q\right)$ is the set of all $n×n$ matrices over the field with $q$ elements that respect a non-singular quadratic form. The value of $d$ must be $0$ for odd $n$, or $1$ or $-1$ for even $n$.
 • The GeneralOrthogonalGroup( d, n, q ) command returns a permutation group isomorphic to the general orthogonal group $GO\left(d,n,q\right)$ for the implemented values of d, n and q.
 • The implemented ranges for n and q are as follows:

 $n=2$ $q\le 100$ $n=3$ $q\le 20$ $n=4$ $q\le 10$ $n=5$ $q\le 5$ $n=6,7,8$ $q=3$ $n=9,10,11$ $q=2$

Examples

 > with(GroupTheory):
 > G := GeneralOrthogonalGroup( 0, 7, 2 );
 ${G}{:=}{\mathbf{GO}}\left({7}{,}{2}\right)$ (1)
 > Generators( G );
 $\left[\left({5}{,}{6}\right)\left({7}{,}{9}\right)\left({8}{,}{11}\right)\left({10}{,}{14}\right)\left({12}{,}{16}\right)\left({13}{,}{18}\right)\left({15}{,}{21}\right)\left({17}{,}{23}\right)\left({19}{,}{26}\right)\left({22}{,}{29}\right)\left({24}{,}{31}\right)\left({25}{,}{33}\right)\left({27}{,}{36}\right)\left({30}{,}{39}\right)\left({34}{,}{43}\right)\left({35}{,}{45}\right)\left({38}{,}{47}\right)\left({40}{,}{49}\right)\left({42}{,}{52}\right)\left({44}{,}{48}\right)\left({50}{,}{54}\right)\left({51}{,}{58}\right)\left({55}{,}{61}\right)\left({56}{,}{57}\right){,}\left({2}{,}{3}{,}{4}{,}{5}{,}{7}{,}{10}\right)\left({6}{,}{8}{,}{12}{,}{17}{,}{24}{,}{32}\right)\left({9}{,}{13}{,}{19}{,}{27}{,}{37}{,}{47}\right)\left({11}{,}{15}{,}{14}{,}{20}{,}{28}{,}{38}\right)\left({16}{,}{22}{,}{23}{,}{30}{,}{40}{,}{50}\right)\left({18}{,}{25}{,}{34}{,}{44}{,}{54}{,}{21}\right)\left({26}{,}{35}\right)\left({31}{,}{41}{,}{51}{,}{43}{,}{49}{,}{57}\right)\left({33}{,}{42}{,}{53}{,}{60}{,}{61}{,}{45}\right)\left({36}{,}{46}{,}{55}\right)\left({39}{,}{48}{,}{56}\right)\left({52}{,}{59}{,}{62}{,}{63}{,}{64}{,}{58}\right)\right]$ (2)
 > G := GeneralOrthogonalGroup( 1, 4, 5 );
 ${G}{:=}{\mathbf{GO}}\left({4}{,}{5}\right)$ (3)
 > GroupOrder( G );
 ${28800}$ (4)
 > G := GeneralOrthogonalGroup( -1, 4, 5 );
 ${G}{:=}{\mathbf{GO}}\left({4}{,}{5}\right)$ (5)
 > Degree( G );
 ${104}$ (6)
 > GroupOrder( G );
 ${31200}$ (7)
 > GroupOrder( GeneralOrthogonalGroup( 0, 7, 3 ) );
 ${18341406720}$ (8)
 > GroupOrder( GeneralOrthogonalGroup( 1, 8, 2 ) );
 ${348364800}$ (9)
 > GroupOrder( GeneralOrthogonalGroup( -1, 8, 2 ) );
 ${394813440}$ (10)

Compatibility

 • The GroupTheory[GeneralOrthogonalGroup] command was introduced in Maple 17.