SymplecticGroup - Maple Help

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GroupTheory

 SymplecticGroup
 construct a permutation group isomorphic to a symplectic group

 Calling Sequence SymplecticGroup(n, q)

Parameters

 n - an even positive integer q - power of a prime number

Description

 • The symplectic group $Sp\left(n,q\right)$ is the group of all $n×n$ matrices over the field with $q$ elements that respect a fixed nondegenerate symplectic form. The integer $n$ must be even.
 • The SymplecticGroup( n, q ) command returns a permutation group isomorphic to the symplectic group $Sp\left(n,q\right)$ .
 • Note that for $n=2$ the groups $Sp\left(n,q\right)$ and $SL\left(n,q\right)$ are isomorphic, so that a special linear group is returned in this case.
 • The implemented ranges for $n$ and $q$ are as follows:

 $n=2$ $q\le 100$ $n=4$ $q\le 10$ $n=6$ $q\le 4$ $n=8,10$ $q=2$

 • If either, or both, of n and q is non-numeric, then a symbolic group representing the symplectic group is returned.
 • In the Standard Worksheet interface, you can insert this group into a document or worksheet by using the Group Constructors palette.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SymplecticGroup}\left(4,5\right)$
 ${G}{≔}{\mathbf{Sp}}\left({4}{,}{5}\right)$ (1)
 > $\mathrm{ifactor}\left(\mathrm{GroupOrder}\left(G\right)\right)$
 ${\left({2}\right)}^{{7}}{}{\left({3}\right)}^{{2}}{}{\left({5}\right)}^{{4}}{}\left({13}\right)$ (2)
 > $\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(2,G\right)\right)$
 ${128}$ (3)
 > $\mathrm{S3}≔\mathrm{SylowSubgroup}\left(3,G\right)$
 ${\mathrm{S3}}{≔}{\mathrm{}}$ (4)
 > $\mathrm{GroupOrder}\left(\mathrm{S3}\right)$
 ${9}$ (5)
 > $\mathrm{IsCyclic}\left(\mathrm{S3}\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{IdentifySmallGroup}\left(\mathrm{S3}\right)$
 ${9}{,}{2}$ (7)
 > $\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(5,G\right)\right)$
 ${625}$ (8)
 > $\mathrm{IsTrivial}\left(\mathrm{PCore}\left(5,G\right)\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{GroupOrder}\left(\mathrm{SylowSubgroup}\left(13,G\right)\right)$
 ${13}$ (10)
 > $G≔\mathrm{SymplecticGroup}\left(4,3\right)$
 ${G}{≔}{\mathbf{Sp}}\left({4}{,}{3}\right)$ (11)
 > $\mathrm{Degree}\left(G\right)$
 ${80}$ (12)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{false}}$ (13)
 > $\mathrm{GroupOrder}\left(\mathrm{Centre}\left(G\right)\right)$
 ${2}$ (14)

For $n=2$ the corresponding special linear group is returned.

 > $\mathrm{SymplecticGroup}\left(2,5\right)$
 ${\mathbf{SL}}\left({2}{,}{5}\right)$ (15)

Note the exceptional isomorphism:

 > $\mathrm{AreIsomorphic}\left(\mathrm{SymplecticGroup}\left(4,2\right),\mathrm{Symm}\left(6\right)\right)$
 ${\mathrm{true}}$ (16)
 > $G≔\mathrm{SymplecticGroup}\left(6,q\right)$
 ${G}{≔}{\mathbf{Sp}}\left({6}{,}{q}\right)$ (17)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${{q}}^{{9}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{-}{1}\right){}\left({{q}}^{{6}}{-}{1}\right)$ (18)
 > $\mathrm{ClassNumber}\left(\mathrm{SymplecticGroup}\left(8,q\right)\right)$
 $\left\{\begin{array}{cc}{5}{}{q}{+}\left({q}{+}{1}\right){}{q}{+}{4}{}{{q}}^{{2}}{+}\left({{q}}^{{2}}{+}{q}{+}{3}\right){}{q}{+}{{q}}^{{4}}{+}{{q}}^{{3}}{+}{7}& {q}{::}{\mathrm{even}}\\ {25}{}{q}{+}{51}{+}\left({q}{+}{4}\right){}{q}{+}{11}{}{{q}}^{{2}}{+}\left({{q}}^{{2}}{+}{4}{}{q}{+}{10}\right){}{q}{+}{{q}}^{{4}}{+}{4}{}{{q}}^{{3}}& {\mathrm{otherwise}}\end{array}\right\$ (19)
 > $\mathrm{ClassNumber}\left(\mathrm{SymplecticGroup}\left(4,{11}^{k}\right)\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\mathbf{assuming}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}k::'\mathrm{posint}'$
 ${5}{}{{11}}^{{k}}{+}{10}{+}{\left({{11}}^{{k}}\right)}^{{2}}$ (20)

Compatibility

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