Conjugation (group theory) in the context of "Binary operation"

In mathematics, especially group theory, the centralizer (also called commutant) of a subset S in a group G is the set ${\displaystyle \operatorname {C} _{G}(S)}$ of elements of G that commute with every element of S, or equivalently, the set of elements ${\displaystyle g\in G}$ such that conjugation by ${\displaystyle g}$ leaves each element of S fixed. The normalizer of S in G is the set of elements ${\displaystyle \mathrm {N} _{G}(S)}$ of G that satisfy the weaker condition of leaving the set ${\displaystyle S\subseteq G}$ 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.

Suitably formulated, the definitions also apply to semigroups.