Centraliser

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 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.