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 Stabilizer( L, G ) command, where L is a list of points in the domain of the permutation group G, computes the iterated stabilizer of L in G. This is the set of elements of G that fix each point in the list L.

The Stabilizer( S, G ) command, where S is a subset of the domain of the permutation group G, computes the set-wise stabilizer of S in G. This is the set of elements $g$ in $G$ that map the set $S$ to itself, but do not necessarily fix each member of $S$.

The Stabiliser command is provided as an alias.

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

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

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

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

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

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

$4$

$G≔\mathrm{SL}\left(3\,3\right)$

$G≔\mathbf{SL}\left(3\,3\right)$

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

$S≔⟨\left(2\,11\,6\,4\,12\,7\right)\left(3\,13\,5\right)\left(8\,10\right)\,\left(5\,7\right)\left(8\,13\right)\left(9\,12\right)\left(10\,11\right)⟩$

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

$432$

$\mathrm{IsSubgroup}\left(S\,G\right)$

$\mathrm{true}$

$\mathrm{IsNormal}\left(S\,G\right)$

$\mathrm{false}$

$S≔\mathrm{Stabilizer}\left(\left[1\,7\,3\,11\right]\,G\right)$

$S≔⟨⟩$

$S≔\mathrm{Stabilizer}\left(\left[1\,2\right]\,G\right)$

$S≔⟨\left(5\,8\,11\right)\left(6\,9\,12\right)\left(7\,10\,13\right)\,\left(5\,10\,12\right)\left(6\,8\,13\right)\left(7\,9\,11\right)\,\left(3\,4\right)\left(6\,7\right)\left(9\,10\right)\left(12\,13\right)\,\left(6\,7\right)\left(8\,11\right)\left(9\,13\right)\left(10\,12\right)⟩$

$\mathrm{AreIsomorphic}\left(S\,\mathrm{DirectProduct}\left(\mathrm{Symm}\left(3\right)\,\mathrm{Symm}\left(3\right)\right)\right)$

$\mathrm{true}$

$S≔\mathrm{Stabilizer}\left(\left\{1\,2\right\}\,G\right)$

$S≔⟨\left(5\,11\,8\right)\left(6\,12\,9\right)\left(7\,13\,10\right)\,\left(5\,10\,12\right)\left(6\,8\,13\right)\left(7\,9\,11\right)\,\left(6\,7\right)\left(8\,11\right)\left(9\,13\right)\left(10\,12\right)\,\left(3\,4\right)\left(8\,11\right)\left(9\,12\right)\left(10\,13\right)\,\left(1\,2\right)\left(3\,4\right)\left(5\,12\,7\,9\right)\left(8\,11\,13\,10\right)⟩$

$\mathrm{AreIsomorphic}\left(S\,\mathrm{WreathProduct}\left(\mathrm{Symm}\left(3\right)\,\mathrm{CyclicGroup}\left(2\right)\right)\right)$

$\mathrm{true}$

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

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