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) = . The center is a normal subgroup, Z(G) ⊲ 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.

As a subgroup

The center of G is always a subgroup of G . In particular: Z(G) contains the identity element of G , because it commutes with every element of g , by definition: eg = g = ge , where e is the identity; x and y are in Z(G) , then so is xy , by associativity: (xy)g = x(yg) = x(gy) = (xg)y = (gx)y = g(xy) for each g ∈ G Z(G) is closed; x is in Z(G) , then so is x−1 as, for all g in G , x−1 commutes with g (gx = xg) ⇒ (x−1gxx−1 = x−1xgx−1) ⇒ (x−1g = gx−1) . 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 G\to Z(G) is not a functor between categories Grp and Ab, since it does not induce a map of arrows.

Conjugacy classes and centralizers

By definition, an element is central whenever its conjugacy class contains only the element itself; i.e. Cl(g) = {g} . The center is the intersection of all the centralizers of elements of G

Conjugation

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 f(g)(h) = ϕg(h) = ghg−1 . 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, G/Z(G) ≃ Inn(G) . The cokernel of this map is the group Out(G) of outer automorphisms, and these form the exact sequence 1 ⟶ Z(G) ⟶ G ⟶ Aut(G) ⟶ Out(G) ⟶ 1 .

Examples

G , is all of G . H , is the set of matrices of the form: Dn , is trivial for odd n ≥ 3 . For even n ≥ 4 , the center consists of the identity element together with the 180° rotation of the polygon. Q8 = {1, −1, i, −i, j, −j, k, −k} , is {1, −1} . Sn , is trivial for n ≥ 3 . An , is trivial for n ≥ 4 . F , GLn(F) , is the collection of scalar matrices, . On(F) is {In, −In} . SO(n) is the whole group when n = 2 , and otherwise when n is even, and trivial when n is odd. G/Z(G) is cyclic, G is abelian (and hence G = Z(G) , so G/Z(G) is trivial).

Higher centers

Quotienting out by the center of a group yields a sequence of groups called the upper central series: (G0 = G) ⟶ (G1 = G0/Z(G0)) ⟶ (G2 = G1/Z(G1)) ⟶ ⋯ The kernel of the map G → Gi is the ''' i th center''' of G (second center, third center, etc.), denoted Zi(G) . Concretely, the ( i+1 )-st center comprises the elements that commute with all elements up to an element of the i th center. Following this definition, one can define the 0th center of a group to be the identity subgroup. This can be continued to transfinite ordinals by transfinite induction; the union of all the higher centers is called the hypercenter. The ascending chain of subgroups 1 ≤ Z(G) ≤ Z2(G) ≤ ⋯ stabilizes at i (equivalently, Zi(G) = Zi+1(G) ) if and only if Gi is centerless.

Examples

Z0(G) = Z1(G) of stabilization. Z1(G) = Z2(G) .