Definition:Center (Abstract Algebra)/Group

This page is about Center in the context of Group Theory. For other uses, see Center.

Definition

The center of a group $G$, denoted $\map Z G$, is the subset of elements in $G$ that commute with every element in $G$.

Symbolically:

$\map Z G = \map {C_G} G = \set {g \in G: g x = x g, \forall x \in G}$

That is, the center of $G$ is the centralizer of $G$ in $G$ itself.

Also known as

Some sources use $Z_G$ to denote this concept.

Some sources use $\map \xi G$.

Also see

• Results about centers of groups can be found here.

Historical Note

The notation $\map Z S$, conventionally used for the center of a structure $\struct {S, \circ}$, derives from the German Zentrum, meaning center.

Linguistic Note

The British English spelling of center is centre.

The convention on $\mathsf{Pr} \infty \mathsf{fWiki}$ is to use the American English spelling center, but it is appreciated that there may be lapses.