Center - Maple Help

GroupTheory

 Center
 construct the center of a group

 Calling Sequence Center( G ) Centre( G )

Parameters

 G - a permutation group

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{GroupTheory}\right):$
 > $G≔\mathrm{PermutationGroup}\left(\left\{\left[\left[1,2\right]\right],\left[\left[1,2,3\right],\left[4,5\right]\right]\right\}\right)$
 ${G}{≔}⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩$ (1)
 > $\mathrm{Center}\left(G\right)$
 ${Z}{}\left(⟨\left({1}{,}{2}\right){,}\left({1}{,}{2}{,}{3}\right)\left({4}{,}{5}\right)⟩\right)$ (2)
 > $\mathrm{Center}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$
 ${Z}{}\left({{\mathbf{A}}}_{{4}}\right)$ (3)
 > $G≔\mathrm{MetacyclicGroup}\left(3,4,2\right)$
 ${G}{≔}⟨\left({1}{,}{2}{,}{3}\right)\left({4}{,}{6}{,}{8}\right)\left({5}{,}{7}{,}{9}\right)\left({10}{,}{11}{,}{12}\right){,}\left({1}{,}{4}{,}{10}{,}{5}\right)\left({2}{,}{6}{,}{11}{,}{7}\right)\left({3}{,}{8}{,}{12}{,}{9}\right)⟩$ (4)
 > $\mathrm{IsAbelian}\left(\mathrm{Center}\left(G\right)\right)$
 ${\mathrm{true}}$ (5)
 > $\mathrm{GroupOrder}\left(\mathrm{Center}\left(G\right)\right)$
 ${12}$ (6)
 > $\mathrm{IsNormal}\left(\mathrm{Center}\left(G\right),G\right)$
 ${\mathrm{true}}$ (7)

Compatibility

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