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.

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

$G≔\mathrm{DihedralGroup}\left(6\right)$

$G≔{\mathbf{D}}_6$

$Z≔\mathrm{Center}\left(G\right)$

$Z≔Z\left({\mathbf{D}}_6\right)$

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

$2$

$G≔\mathrm{DihedralGroup}\left(7\right)$

$G≔{\mathbf{D}}_7$

$Z≔\mathrm{Center}\left(G\right)$

$Z≔Z\left({\mathbf{D}}_7\right)$

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

$1$

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

$G≔\mathrm{GL}\left(3\,3\right)$

$G≔\mathbf{GL}\left(3\,3\right)$

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

$\mathrm{true}$

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

$2$

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

$\mathrm{true}$

$G≔\mathrm{FrobeniusGroup}\left(72\,2\right)$

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

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

$1$

$\mathrm{Centre}\left(\mathrm{McLaughlinGroup}\left(\right)\right)$

$\mathrm{Centre}\left(\mathrm{CyclicGroup}\left(24\right)\right)$

$C_{24}$

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

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