# Centre of a group

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

Formally,

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

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.