Centralizer and normalizer

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

Source: Wikipedia — Centralizer and normalizer (CC BY-SA 4.0)