The centralizer of an element z of a group G is the set of elements of G which commute with z, C_G(z) = {x element G, x z = z x}. Likewise, the centralizer of a subgroup H of a group G is the set of elements of G which commute with every element of H, C_G(H) = {x element G, for all h element H, x h = h x}. The centralizer always contains the group center of the group and is contained in the corresponding normalizer. In an Abelian group, the centralizer is the whole group.

Abelian group | group | group center | normalizer | subgroup