Centralizer - Maple Help

GroupTheory

 Centralizer
 construct the centralizer of an element of a group

 Calling Sequence Centralizer( g, G ) Centraliser( g, G )

Parameters

 G - a permutation group or a Cayley table group g - an element of G

Description

 • The centralizer of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$. That is, an element $c$ of $G$ belongs to the centralizer of $g$ if, and only if, $g·c=c·g$.
 • The Centralizer( g, G ) command constructs the centralizer of the element $g$ of a group G. The group G must be an instance of a permutation group, a group defined by a Cayley table, or a custom group that defines its own centralizer method.
 • The centralizer of $g$ in $G$ may also be thought of as the stabilizer of $g$ under the action of $G$ on itself by conjugation.
 • The Centraliser command is provided as an alias.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1,2\right]\right]\right),\mathrm{Perm}\left(\left[\left[1,2,3\right],\left[4,5\right]\right]\right)\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (1)
 > $C≔\mathrm{Centralizer}\left(\mathrm{Perm}\left(\left[\left[1,2,3\right]\right]\right),G\right)$
 ${C}{≔}⟨\left({1}{,}{3}{,}{2}\right)\left({4}{,}{5}\right){,}\left({1}{,}{2}{,}{3}\right)⟩$ (2)
 > $\mathrm{Generators}\left(C\right)$
 $\left[\left({1}{,}{3}{,}{2}\right)\left({4}{,}{5}\right){,}\left({1}{,}{2}{,}{3}\right)\right]$ (3)
 > $\mathrm{GroupOrder}\left(C\right)$
 ${6}$ (4)

Compatibility

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