Stabilizer - Maple Help

GroupTheory

 Stabilizer
 construct the stabilizer of a point in a permutation group

 Calling Sequence Stabilizer( alpha, G ) Stabiliser( alpha, G )

Parameters

 G - a permutation group alpha - posint; the point whose stabilizer is to be computed

Description

 • The stabilizer of a point $\mathrm{\alpha }$ under a permutation group $G$ is the set of elements of $G$ that fix $\mathrm{\alpha }$.  It is a subgroup of $G$. That is, an element $g$ in $G$ belongs to the stabilizer of $\mathrm{\alpha }$ if ${\mathrm{\alpha }}^{g}=\mathrm{\alpha }$.
 • The Stabilizer( alpha, G ) command computes the stabilizer of the point alpha under the action of the permutation group G.
 • The Stabiliser command is provided as an alias.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Group}\left(\left\{\left[\left[1,2\right]\right],\left[\left[4,5\right]\right]\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({4}{,}{5}\right)⟩$ (1)
 > $S≔\mathrm{Stabilizer}\left(3,G\right)$
 ${S}{≔}⟨\left({1}{,}{2}\right){,}\left({4}{,}{5}\right)⟩$ (2)
 > $\mathrm{GroupOrder}\left(S\right)$
 ${4}$ (3)
 > $G≔\mathrm{SL}\left(3,3\right)$
 ${G}{≔}{\mathbf{SL}}\left({3}{,}{3}\right)$ (4)
 > $S≔\mathrm{Stabilizer}\left(1,G\right)$
 ${S}{≔}{\mathrm{< a permutation group on 13 letters with 8 generators >}}$ (5)
 > $\mathrm{GroupOrder}\left(S\right)$
 ${432}$ (6)
 > $\mathrm{IsSubgroup}\left(S,G\right)$
 ${\mathrm{true}}$ (7)
 > $\mathrm{IsNormal}\left(S,G\right)$
 ${\mathrm{false}}$ (8)

Compatibility

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