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.

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

$G≔\mathrm{Group}\left(\mathrm{Perm}\left(\left[\left[1\,2\right]\right]\right)\,\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\,\left[4\,5\right]\right]\right)\right)$

$G≔⟨\left(1\,2\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩$

$C≔\mathrm{Centralizer}\left(\mathrm{Perm}\left(\left[\left[1\,2\,3\right]\right]\right)\,G\right)$

$C≔⟨\left(4\,5\right)\,\left(1\,2\,3\right)\,\left(1\,2\,3\right)\left(4\,5\right)⟩$

$\mathrm{Generators}\left(C\right)$

$\left[\left(4\,5\right)\,\left(1\,2\,3\right)\,\left(1\,2\,3\right)\left(4\,5\right)\right]$

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

$6$

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

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