Definition:Center (Abstract Algebra)/Group

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

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.

Also see

 * Equivalence of Definitions of Abelian Group
 * Center of Group is Abelian Subgroup
 * Center of Group is Normal Subgroup: $Z \paren G \lhd G$ for any group $G$.
 * Center is Intersection of Centralizers