Centraliser

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In group theory, the centraliser of a subset of a group (mathematics) is the set of all group elements which commute with every element of the given subset.

Formally, for S a subset of a group G, we define

C_G(S) = \{ g \in G : \forall s \in S,~ gs=sg \} . \,

The centraliser of any set is a subgroup of G, and the centraliser of S is equal to the centraliser of the subgroup \langle S \rangle generated by the subset S.

The centraliser of the empty set is the whole group G; the centraliser of the whole group G is the centre of G.

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