Remark

(1) The normalizer (resp. centralizer) of A is the strict stabilizer (resp. fixer) of A when G operates on itself by inner automorphisms. In particular the centralizer is a normal subgroup of the normalizer.

(2) The set of elements $b \in G$ such that $bAb^{-1} \subset A$ is a submonoid of G. Even when A is a subgroup of G, this set is not necessarily a subgroup of G (Exercise 27).