Center (group theory)

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, 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, Z(G) ⊲ G. As a subgroup, it is always characteristic, 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 sometimes called central. The center of G is always a subgroup of G. In particular: Furthermore, the center of G is always a normal subgroup of G. Since all elements of Z(G) commute, it is closed under conjugation. By definition, the center is the set of elements for which the conjugacy class of each element is the element itself; i.e., Cl(g) = {g}. The center is also the intersection of all the centralizers of each element of G. As centralizers are subgroups, this again shows that the center is a subgroup. Consider the map, f: G → Aut(G), from G to the automorphism group of G defined by f(g) = ϕg, where ϕg is the automorphism of G defined by