Center (group theory)

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Cayley table of Dih4
The center is {0,7}: The row starting with 7 is the transpose of the column starting with 7. The entries 7 are symmetric to the main diagonal. (Only for the neutral element this is granted in all groups.)

In abstract algebra, the center of a group G, denoted Z(G),[note 1] is the set of elements that commute with every element of G. In set-builder notation,

Z(G) = \{z \in G \mid \forall g\in G, zg = gz \}.

The center is a subgroup of G, which by definition is abelian (that is commutative). As a subgroup, it is always normal, and indeed characteristic, but it need not be fully characteristic. The quotient group G / Z(G) is isomorphic to the group of inner automorphisms of 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[edit]

The center of G is always a subgroup of G. In particular:

  1. Z(G) contains e, the identity element of G, because eg = g = ge for all g ∈ G by definition of e, so by definition of Z(G), eZ(G);
  2. If x and y are in Z(G), then (xy)g = x(yg) = x(gy) = (xg)y = (gx)y = g(xy) for each gG, and so xy is in Z(G) as well (i.e., Z(G) exhibits closure);
  3. If x is in Z(G), then gx = xg, and multiplying twice, once on the left and once on the right, by x−1, gives x−1g = gx−1 — so x−1Z(G).

Furthermore the center of G is always a normal subgroup of G, as it is closed under conjugation.

Conjugacy classes and centralisers[edit]

By definition, the center is the set of elements for which the conjugacy class of each element is the element itself, i.e. ccl(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[edit]

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

\phi_g(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)\cong \rm{Inn}(G).

The cokernel of this map is the group \operatorname{Out}(G) of outer automorphisms, and these form the exact sequence

1 \to Z(G) \to G \to \operatorname{Aut}(G) \to \operatorname{Out}(G) \to 1.

Examples[edit]

Higher centers[edit]

Quotienting out by the center of a group yields a sequence of groups called the upper central series:

G_0 = G \to G_1 = G_0/Z(G_0) \to G_2 = G_1/Z(G_1) \to \cdots

The kernel of the map G \to G_i is the ith center of G (second center, third center, etc.), and is denoted Z^i(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.[note 2]

The ascending chain of subgroups

1 \leq Z(G) \leq Z^2(G) \leq \cdots

stabilizes at i (equivalently, Z^i(G) = Z^{i+1}(G)) if and only if G_i is centerless.

Examples[edit]

  • For a centerless group, all higher centers are zero, which is the case Z^0(G)=Z^1(G) of stabilization.
  • By Grün's lemma, the quotient of a perfect group by its center is centerless, hence all higher centers equal the center. This is a case of stabilization at Z^1(G)=Z^2(G).

See also[edit]

Notes[edit]

  1. ^ The notation Z is from German Zentrum, meaning "center".
  2. ^ This union will include transfinite terms if the UCS does not stabilize at a finite stage.

External links[edit]