Centralizer and normalizer

Summary

In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set $\operatorname{C}_G(S)$ of elements of G that commute with every element of S, or equivalently, the set of elements $g\in G$ such that conjugation by $g$ leaves each element of S fixed. The normalizer of S in G is the set of elements $\mathrm{N}_G(S)$ of G that satisfy the weaker condition of leaving the set $S \subseteq G$ fixed under conjugation. The centralizer and normalizer of S are subgroups of G. Many techniques in group theory are based on studying the centralizers and normalizers of suitable subsets S.