Stabilizer - Maple Help

GroupTheory

 Stabilizer
 construct the stabilizer of a point, list, or set in a permutation group

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

Parameters

 G - a permutation group alpha - posint; the point whose stabilizer is to be computed L - list(posint); a list of points S - set(posint); a set of points

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 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.

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)
 > $S≔\mathrm{Stabilizer}\left(3,G\right)$
 ${S}{≔}⟨\left({4}{,}{5}\right){,}\left({1}{,}{2}\right)\left({4}{,}{5}\right)⟩$ (2)
 > $\mathrm{GroupOrder}\left(S\right)$
 ${4}$ (3)
 > $G≔\mathrm{SL}\left(3,3\right)$
 ${G}{≔}{\mathbf{SL}}\left({3}{,}{3}\right)$ (4)
 > $S≔\mathrm{Stabilizer}\left(1,G\right)$
 ${S}{≔}⟨\left({2}{,}{11}{,}{6}{,}{4}{,}{12}{,}{7}\right)\left({3}{,}{13}{,}{5}\right)\left({8}{,}{10}\right){,}\left({5}{,}{7}\right)\left({8}{,}{13}\right)\left({9}{,}{12}\right)\left({10}{,}{11}\right)⟩$ (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)
 > $S≔\mathrm{Stabilizer}\left(\left[1,7,3,11\right],G\right)$
 ${S}{≔}⟨⟩$ (9)
 > $S≔\mathrm{Stabilizer}\left(\left[1,2\right],G\right)$
 ${S}{≔}{\mathrm{< a permutation group on 13 letters with 4 generators >}}$ (10)
 > $\mathrm{AreIsomorphic}\left(S,\mathrm{DirectProduct}\left(\mathrm{Symm}\left(3\right),\mathrm{Symm}\left(3\right)\right)\right)$
 ${\mathrm{true}}$ (11)
 > $S≔\mathrm{Stabilizer}\left(\left\{1,2\right\},G\right)$
 ${S}{≔}⟨\left({1}{,}{2}\right)\left({5}{,}{8}{,}{10}{,}{13}{,}{12}{,}{6}\right)\left({7}{,}{11}{,}{9}\right){,}\left({3}{,}{4}\right)\left({8}{,}{11}\right)\left({9}{,}{12}\right)\left({10}{,}{13}\right){,}\left({5}{,}{8}{,}{11}\right)\left({6}{,}{9}{,}{12}\right)\left({7}{,}{10}{,}{13}\right)⟩$ (12)
 > $\mathrm{AreIsomorphic}\left(S,\mathrm{WreathProduct}\left(\mathrm{Symm}\left(3\right),\mathrm{CyclicGroup}\left(2\right)\right)\right)$
 ${\mathrm{true}}$ (13)

Compatibility

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