GroupTheory/SymplecticSemilinearGroup - Maple Help

GroupTheory

 SymplecticSemilinearGroup
 construct a permutation group isomorphic to the symplectic semi-linear group over a finite field

 Calling Sequence SymplecticSemilinearGroup(n, q) Sigmap( n, q )

Parameters

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

Description

 • The symplectic semi-linear group $\Sigma p\left(n,q\right)$ is the set of all semi-linear transformations of an $n$-dimensional vector space $V$ over the field with $q$ elements whose linear part preserves a non-degenerate symplectic form. The dimension $n$ must be an even positive integer. The group $\Sigma p\left(n,q\right)$ is a semi-direct product of the symplectic group $Sp\left(n,q\right)$ with the Galois group of the field GF(q). Therefore, if $q$ is prime, $\Sigma p\left(n,q\right)$ is isomorphic to $Sp\left(n,q\right)$ . Furthermore, if $n=2$, then $Sp\left(n,q\right)$ and $SL\left(n,q\right)$ coincide, so $\Sigma L\left(2,q\right)$ is returned in this case.
 • If n and q are positive integers, then the SymplecticSemilinearGroup( n, q ) command returns a permutation group isomorphic to the symplectic semi-linear group  $\Sigma p\left(n,q\right)$ . Otherwise, a symbolic group is returned, with which Maple can do some limited computations.
 • The abbreviation Sigmap( n, q ) is available as a synonym for SymplecticSemilinearGroup( n, q ).

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{SymplecticSemilinearGroup}\left(2,8\right)$
 ${G}{≔}{\mathbf{\Sigma L}}\left({2}{,}{8}\right)$ (1)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${1512}$ (2)

Notice that

 > $\mathrm{GroupOrder}\left(\mathrm{SigmaL}\left(2,8\right)\right)$
 ${1512}$ (3)
 > $G≔\mathrm{Sigmap}\left(4,4\right)$
 ${G}{≔}{\mathbf{\Sigma p}}\left({4}{,}{4}\right)$ (4)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${1958400}$ (5)
 > $\mathrm{IsSimple}\left(G\right)$
 ${\mathrm{false}}$ (6)
 > $\mathrm{IsSimple}\left(\mathrm{DerivedSubgroup}\left(G\right)\right)$
 ${\mathrm{true}}$ (7)
 > $G≔\mathrm{Sigmap}\left(8,q\right)$
 ${G}{≔}{\mathbf{\Sigma p}}\left({8}{,}{q}\right)$ (8)
 > $\mathrm{GroupOrder}\left(G\right)$
 ${\mathrm{logp}}{}\left({q}\right){}{{q}}^{{16}}{}\left({{q}}^{{2}}{-}{1}\right){}\left({{q}}^{{4}}{-}{1}\right){}\left({{q}}^{{6}}{-}{1}\right){}\left({{q}}^{{8}}{-}{1}\right)$ (9)