Center

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, Z(G) = {z ∈ G ∣ ∀g ∈ G, zg = gz} 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.