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.

$\mathrm{with}\left(\mathrm{GroupTheory}\right)\:$

$G≔\mathrm{Group}\left(\left\{\left[\left[1\,2\right]\right]\,\left[\left[4\,5\right]\right]\right\}\right)$

$G≔⟨\left(1\,2\right)\,\left(4\,5\right)⟩$

$S≔\mathrm{Stabilizer}\left(3\,G\right)$

$S≔⟨\left(1\,2\right)\,\left(4\,5\right)⟩$

$\mathrm{GroupOrder}\left(S\right)$

$4$

$G≔\mathrm{SL}\left(3\,3\right)$

$G≔\mathbf{SL}\left(3\,3\right)$

$S≔\mathrm{Stabilizer}\left(1\,G\right)$

$S≔\mathrm{<\; a\; permutation\; group\; on\; 13\; letters\; with\; 8\; generators\; >}$

$\mathrm{GroupOrder}\left(S\right)$

$432$

$\mathrm{IsSubgroup}\left(S\&comma;G\right)$

$\mathrm{true}$

$\mathrm{IsNormal}\left(S\&comma;G\right)$

$\mathrm{false}$

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

For more information on Maple 17 changes, see Updates in Maple 17.