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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,
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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.
As a subgroup
The center of G is always a subgroup of G. In particular:
- Z(G) contains the identity element of G, e, because, for all g ∈ G, it commutes with g, eg = g = ge, by definition;
- If 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; i.e., Z(G) is closed;
- If 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 a normal subgroup of G, as it is closed under conjugation, as all elements commute.
Conjugacy classes and centralisers
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.
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.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 ith center of G (second center, third center, etc.), and is denoted Zi(G). Concretely, the (i + 1)-st center are the terms that commute with all elements up to an element of the ith 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.