# Centraliser

From Citizendium

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

The centraliser of any set is a subgroup of *G*, and the centraliser of *S* is equal to the centraliser of the subgroup 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*.