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

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

The centraliser of any set is a subgroup of G, and the centraliser of S is equal to the centraliser of the subgroup ${\displaystyle ⟨S⟩}$ 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.