Center (group theory)

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) = {z \in G | \forall g \in G, zg = gz}

.

Cayley table for D₄ showing elements of the center, {e, a²}, commute with all other elements (this can be seen by noticing that all occurrences of a given center element are arranged symmetrically about the center diagonal or by noticing that the row and column starting with a given center element are transposes of each other).
∘	e	b	a	a²	a³	ab	a²b	a³b
e	e	b	a	a²	a³	ab	a²b	a³b
b	b	e	a³b	a²b	ab	a³	a²	a
a	a	ab	a²	a³	e	a²b	a³b	b
a²	a²	a²b	a³	e	a	a³b	b	ab
a³	a³	a³b	e	a	a²	b	ab	a²b
ab	ab	a	b	a³b	a²b	e	a³	a²
a²b	a²b	a²	ab	b	a³b	a	e	a³
a³b	a³b	a³	a²b	ab	b	a²	a	e

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.

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