Center (group theory)

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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,

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.

Contents

As a subgroup

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

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

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

This 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 \operatorname{Out}(G) of outer automorphisms, and these form the exact sequence

Examples

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