Definition:Center (Abstract Algebra)/Group

This page is about the center of a Group.

For the center of a ring, see Center of a Ring.

For the center of a circle, see Center of a Circle.

Definition
The center of a group $G$, denoted $Z \left({G}\right)$, is the subset of elements in $G$ that commute with every element in $G$.

Symbolically:
 * $Z \left({G}\right) = C_G \left({G}\right) = \left\{{g \in G: g x = x g, \forall x \in G}\right\}$

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

It is shown here that $Z \left({G}\right) \triangleleft G$ for any group $G$.