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 $$G$$.