GeneralOrthogonalGroup - Maple Help

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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:

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

 • If the argument q is not a prime power (and is non-numeric), then a symbolic group representing $GO\left(d,n,q\right)$ is returned.
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{GeneralOrthogonalGroup}\left(0,7,2\right)$
 ${G}{≔}{\mathbf{GO}}\left({7}{,}{2}\right)$ (1)
 > $\mathrm{Generators}\left(G\right)$
 $\left[\left({1}{,}{2}{,}{3}{,}{4}{,}{5}{,}{7}\right)\left({6}{,}{8}{,}{11}{,}{15}{,}{21}{,}{27}\right)\left({9}{,}{12}{,}{16}{,}{22}{,}{29}{,}{38}\right)\left({10}{,}{13}{,}{18}{,}{23}{,}{31}{,}{41}\right)\left({14}{,}{19}{,}{24}{,}{33}{,}{44}{,}{34}\right)\left({17}{,}{20}{,}{25}\right)\left({26}{,}{35}{,}{45}{,}{53}{,}{32}{,}{42}\right)\left({28}{,}{36}{,}{47}\right)\left({30}{,}{39}{,}{50}{,}{56}{,}{61}{,}{58}\right)\left({37}{,}{48}\right)\left({40}{,}{51}{,}{46}{,}{54}{,}{59}{,}{62}\right)\left({49}{,}{55}{,}{60}{,}{63}{,}{52}{,}{57}\right){,}\left({5}{,}{6}\right)\left({7}{,}{9}\right)\left({8}{,}{10}\right)\left({11}{,}{14}\right)\left({13}{,}{17}\right)\left({15}{,}{20}\right)\left({18}{,}{19}\right)\left({21}{,}{26}\right)\left({22}{,}{28}\right)\left({23}{,}{30}\right)\left({24}{,}{32}\right)\left({25}{,}{34}\right)\left({29}{,}{37}\right)\left({31}{,}{40}\right)\left({33}{,}{43}\right)\left({36}{,}{46}\right)\left({38}{,}{41}\right)\left({39}{,}{49}\right)\left({42}{,}{44}\right)\left({45}{,}{52}\right)\left({48}{,}{51}\right)\left({53}{,}{58}\right)\left({57}{,}{62}\right)\left({59}{,}{61}\right)\right]$ (2)
 > $G≔\mathrm{GeneralOrthogonalGroup}\left(1,4,5\right)$
 ${G}{≔}{\mathbf{GO}}\left({4}{,}{5}\right)$ (3)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${28800}$ (4)
 > $G≔\mathrm{GeneralOrthogonalGroup}\left(-1,4,5\right)$
 ${G}{≔}{\mathbf{GO}}\left({4}{,}{5}\right)$ (5)
 > $\mathrm{Degree}\left(G\right)$
 ${104}$ (6)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${31200}$ (7)
 > $G≔\mathrm{GeneralOrthogonalGroup}\left(0,3,5\right)$
 ${G}{≔}{\mathbf{GO}}\left({3}{,}{5}\right)$ (8)
 > $\mathrm{CharacterTable}\left(G\right)$
 $\begin{array}{c}\left[\begin{array}{ccccccccccc}{C}& {\mathrm{1a}}& {\mathrm{2a}}& {\mathrm{2b}}& {\mathrm{2c}}& {\mathrm{2d}}& {\mathrm{2e}}& {\mathrm{3a}}& {\mathrm{4a}}& {\mathrm{4b}}& {\dots }\\ {\mathrm{|C|}}& {1}& {1}& {10}& {10}& {15}& {15}& {20}& {30}& {30}& {\dots }\\ {\mathrm{X1}}& {1}& {1}& {1}& {1}& {1}& {1}& {1}& {1}& {1}& {\dots }\\ {\mathrm{X2}}& {1}& {-1}& {-1}& {1}& {1}& {-1}& {1}& {-1}& {1}& {\dots }\\ {\mathrm{X3}}& {1}& {-1}& {1}& {-1}& {1}& {-1}& {1}& {1}& {-1}& {\dots }\\ {\mathrm{X4}}& {1}& {1}& {-1}& {-1}& {1}& {1}& {1}& {-1}& {-1}& {\dots }\\ {\mathrm{X5}}& {4}& {-4}& {-2}& {2}& {0}& {0}& {1}& {0}& {0}& {\dots }\\ {\mathrm{X6}}& {4}& {-4}& {2}& {-2}& {0}& {0}& {1}& {0}& {0}& {\dots }\\ {\mathrm{X7}}& {4}& {4}& {-2}& {-2}& {0}& {0}& {1}& {0}& {0}& {\dots }\\ {\mathrm{X8}}& {4}& {4}& {2}& {2}& {0}& {0}& {1}& {0}& {0}& {\dots }\\ {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {⋮}& {}\end{array}\right]\\ \hfill {\text{16 × 15 Matrix}}\end{array}$ (9)
 > $\mathrm{OrderClassPolynomial}\left(G,x\right)$
 ${24}{}{{x}}^{{10}}{+}{60}{}{{x}}^{{6}}{+}{24}{}{{x}}^{{5}}{+}{60}{}{{x}}^{{4}}{+}{20}{}{{x}}^{{3}}{+}{51}{}{{x}}^{{2}}{+}{x}$ (10)
 > $\mathrm{DerivedSeries}\left(G\right)$
 ${\mathbf{GO}}\left({3}{,}{5}\right){▹}⟨\left({1}{,}{4}\right)\left({2}{,}{8}\right)\left({3}{,}{9}\right)\left({5}{,}{15}\right)\left({6}{,}{23}\right)\left({7}{,}{20}\right)\left({10}{,}{12}\right)\left({11}{,}{19}\right)\left({13}{,}{16}\right)\left({14}{,}{24}\right)\left({17}{,}{21}\right)\left({18}{,}{22}\right){,}\left({1}{,}{4}\right)\left({2}{,}{8}\right)\left({3}{,}{22}\right)\left({5}{,}{6}\right)\left({7}{,}{18}\right)\left({9}{,}{10}\right)\left({11}{,}{23}\right)\left({12}{,}{17}\right)\left({13}{,}{14}\right)\left({15}{,}{16}\right)\left({19}{,}{24}\right)\left({20}{,}{21}\right){,}\left({1}{,}{20}\right)\left({2}{,}{19}\right)\left({3}{,}{9}\right)\left({4}{,}{21}\right)\left({5}{,}{15}\right)\left({6}{,}{16}\right)\left({7}{,}{12}\right)\left({8}{,}{24}\right)\left({10}{,}{22}\right)\left({11}{,}{13}\right)\left({14}{,}{23}\right)\left({17}{,}{18}\right)⟩$ (11)
 > $\mathrm{Hypercentre}\left(G\right)$
 $⟨\left({1}{,}{4}\right)\left({2}{,}{8}\right)\left({3}{,}{9}\right)\left({5}{,}{15}\right)\left({6}{,}{16}\right)\left({7}{,}{17}\right)\left({10}{,}{22}\right)\left({11}{,}{14}\right)\left({12}{,}{18}\right)\left({13}{,}{23}\right)\left({19}{,}{24}\right)\left({20}{,}{21}\right)⟩$ (12)
 > $\mathrm{IsMalnormal}\left(\mathrm{SylowSubgroup}\left(2,G\right),G\right)$
 ${\mathrm{false}}$ (13)
 > $\mathrm{GroupOrder}\left(\mathrm{PCore}\left(2,G\right)\right)$
 ${2}$ (14)
 > $\mathrm{IsMalnormal}\left(\mathrm{SylowSubgroup}\left(3,G\right),G\right)$
 ${\mathrm{false}}$ (15)
 > $\mathrm{IsMalnormal}\left(\mathrm{SylowSubgroup}\left(5,G\right),G\right)$
 ${\mathrm{false}}$ (16)
 > $\mathrm{GroupOrder}\left(\mathrm{GeneralOrthogonalGroup}\left(0,7,3\right)\right)$
 ${18341406720}$ (17)
 > $\mathrm{GroupOrder}\left(\mathrm{GeneralOrthogonalGroup}\left(1,8,2\right)\right)$
 ${348364800}$ (18)
 > $\mathrm{GroupOrder}\left(\mathrm{GeneralOrthogonalGroup}\left(-1,8,2\right)\right)$
 ${394813440}$ (19)
 > $\mathrm{GroupOrder}\left(\mathrm{GeneralOrthogonalGroup}\left(-1,4,q\right)\right)$
 ${\mathrm{igcd}}{}\left({2}{,}{q}{-}{1}\right){}{\mathrm{igcd}}{}\left({2}{,}{q}\right){}{{q}}^{{2}}{}\left({{q}}^{{2}}{+}{1}\right){}\left({{q}}^{{2}}{-}{1}\right)$ (20)
 > $\mathrm{GroupOrder}\left(\mathrm{GeneralOrthogonalGroup}\left(1,4,q\right)\right)$
 ${\mathrm{igcd}}{}\left({2}{,}{q}{-}{1}\right){}{\mathrm{igcd}}{}\left({2}{,}{q}\right){}{{q}}^{{2}}{}{\left({{q}}^{{2}}{-}{1}\right)}^{{2}}$ (21)

Compatibility

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