Corollary

Let G be a group operating on a set E and A be a subset of E. The strict stabilizer S and the fixer F of A are subgroups of G and F is a normal subgroup of S.

The first assertion follows from the proposition and F is the kernel of the homomorphism of S into $\mathfrak{S}_A$ associated with the operation of S on A.

A group G operates faithfully on a set E if and only if the fixer of E consists of the identity element of G. The fixer of E is the kernel of the given homomorphism of G into $\mathfrak{S}_E$; this homomorphism is injective if and only if its kernel consists of the identity element (§ 4, no. 5, Theorem 3).

Let M be a monoid, E an M-set and $a$ an element of E. The fixer, strict stabilizer and stabilizer of $\{a\}$ are equal; this monoid is called equally the fixer or stabilizer of $a$. The fixer of a subset A of E is the intersection of the fixers of the elements of A. $a$ is called an invariant element of E if the fixer of $a$ is the monoid M. M is said to operate trivially on E if every element of E is invariant.