In abstract algebra, the center of a group G is the set of elements that commute with every element of G. It is denoted Z(G), from German Zentrum, meaning center. In set-builder notation,

The center is a normal subgroup, ${\displaystyle Z(G)\triangleleft G}$, and also a characteristic subgroup, but is not necessarily fully characteristic. The quotient group, G / Z(G), is isomorphic to the inner automorphism group, Inn(G).

A group G is abelian if and only if Z(G) = G. At the other extreme, a group is said to be centerless if Z(G) is trivial; i.e., consists only of the identity element.

The elements of the center are central elements.

The center of G is always a subgroup of G. In particular:

Furthermore, the center of G is always an abelian and normal subgroup of G. Since all elements of Z(G) commute, it is closed under conjugation.

A group homomorphism f : G → H might not restrict to a homomorphism between their centers. The image elements f (g) commute with the image f ( G ), but they need not commute with all of H unless f is surjective. Thus the center mapping ${\displaystyle G\to Z(G)}$ is not a functor between categories Grp and Ab, since it does not induce a map of arrows.

By definition, an element is central whenever its conjugacy class contains only the element itself; i.e. Cl(g) = {g}.