 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

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

Whether the center of a dihedral group is trivial or a group of order two depends upon whether the degree is odd or even.

 > $G≔\mathrm{DihedralGroup}\left(6\right)$
 ${G}{≔}{{\mathbf{D}}}_{{6}}$ (1)
 > $Z≔\mathrm{Center}\left(G\right)$
 ${Z}{≔}{Z}{}\left({{\mathbf{D}}}_{{6}}\right)$ (2)
 > $\mathrm{GroupOrder}\left(Z\right)$
 ${2}$ (3)
 > $G≔\mathrm{DihedralGroup}\left(7\right)$
 ${G}{≔}{{\mathbf{D}}}_{{7}}$ (4)
 > $Z≔\mathrm{Center}\left(G\right)$
 ${Z}{≔}{Z}{}\left({{\mathbf{D}}}_{{7}}\right)$ (5)
 > $\mathrm{GroupOrder}\left(Z\right)$
 ${1}$ (6)
 > $\mathrm{Center}\left(\mathrm{AlternatingGroup}\left(4\right)\right)$
 $⟨⟩$ (7)
 > $G≔\mathrm{GL}\left(3,3\right)$
 ${G}{≔}{\mathbf{GL}}\left({3}{,}{3}\right)$ (8)
 > $\mathrm{IsAbelian}\left(\mathrm{Center}\left(G\right)\right)$
 ${\mathrm{true}}$ (9)
 > $\mathrm{GroupOrder}\left(\mathrm{Center}\left(G\right)\right)$
 ${2}$ (10)
 > $\mathrm{IsNormal}\left(\mathrm{Center}\left(G\right),G\right)$
 ${\mathrm{true}}$ (11)

The center of any Frobenius group is trivial.

 > $G≔\mathrm{FrobeniusGroup}\left(72,2\right)$
 ${G}{≔}{\mathrm{< a permutation group on 9 letters with 5 generators >}}$ (12)
 > $\mathrm{GroupOrder}\left(\mathrm{Centre}\left(G\right)\right)$
 ${1}$ (13)

Likewise, a non-abelian simple group has trivial center.

 > $\mathrm{Centre}\left(\mathrm{McLaughlinGroup}\left(\right)\right)$
 $⟨⟩$ (14)

Of course, every abelian group is equal to its center.

 > $\mathrm{Centre}\left(\mathrm{CyclicGroup}\left(24\right)\right)$
 ${{C}}_{{24}}$ (15) Compatibility

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