Definition:Center (Abstract Algebra)/Group

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.

Also known as
Some sources use $Z_G$ to denote this concept.

Linguistic Note
British English spells this centre.

The convention on is to use the American English spelling center, but it is appreciated that there may be lapses.

Also see

 * Equivalence of Definitions of Abelian Group
 * Center of Group is Abelian Subgroup
 * Center of Group is Normal Subgroup: $Z \left({G}\right) \lhd G$ for any group $G$.
 * Center is Intersection of Centralizers