Centre of a group

In group theory, the centre of a group is the subset of elements which commute with every element of the group.

Formally,

$Z(G)=\{z\in G:\forall x\in G,~xz=zx\}.\,$

The centre is a subgroup, which is normal and indeed characteristic. It may be described as the set of elements by which conjugation is trivial (the identity map); this shows the centre as the kernel of the homomorphism to G to its inner automorphism group.

References

• Marshall Hall jr (1959). The theory of groups. New York: Macmillan, 14.