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.
is not a functor between categories Grp and Ab, since it does not induce a map of arrows.
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 The function, f is a group homomorphism, and its kernel is precisely the center of G, and its image is called the inner automorphism group of G, denoted Inn(G).
By the first isomorphism theorem we get, The cokernel of this map is the group Out(G) of outer automorphisms, and these form the exact sequence Quotienting out by the center of a group yields a sequence of groups called the upper central series: The kernel of the map G → Gi is the ith center[1] of G (second center, third center, etc.
Following this definition, one can define the 0th center of a group to be the identity subgroup.
[note 1] The ascending chain of subgroups stabilizes at i (equivalently, Zi(G) = Zi+1(G)) if and only if Gi is centerless.