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.