The center of a group $G$ is the set of elements of $G$ that commute with all elements of $G$. That is, an element $g$ of $G$ belongs to the center of $G$ if, and only if, $g·x=x·g$, for all $x$ in $G$.

The Center( G ) command constructs the center 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 center method.

The Centre command is provided as an alias.

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

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

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

$\mathrm{Center}\left(G\right)$

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

$\mathrm{Center}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$

$Z\left({\mathbf{A}}_4\right)$

$G≔\mathrm{MetacyclicGroup}\left(3\,4\,2\right)$

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

$\mathrm{IsAbelian}\left(\mathrm{Center}\left(G\right)\right)$

$\mathrm{true}$

$\mathrm{GroupOrder}\left(\mathrm{Center}\left(G\right)\right)$

$12$

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

$\mathrm{true}$

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

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