# Fixer

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A fixer is part of a monoid (or group) acting on a set.

Specifically, let $M$ be a monoid acting on $S$; let $A$ be a subset of $S$. The fixer of $S$ is the set of all $a\in M$ fow which $a(x) = x$ for all $x \in A$. If $S$ contains a single element $x$, we sometimes call this the fixer or stabilizer of $x$, by abuse of language.

Evidently, the fixer of $A$ is a submonoid of $M$ (and of the strict stabilizer of $A$). Also, if $a$ is an invertible element of the fixer of $A$, then $a^{-1}$ is evidently an element of the fixer of $A$.

It follows that if $M$ is a group $G$, then the fixer $F$ of $A$ is a subgroup of $G$. In fact, it is a normal subgroup of the strict stabilizer of $A$. Indeed, it is the kernel of the canonical homomorphism from the strict stabilizer of $A$ to $\mathfrak{S}_A}$ (Error compiling LaTeX. ! Extra }, or forgotten \$.), the group of permutations on $A$.

Note, however, that $F$ need not be a normal subgroup of $G$. For example, if $G$ is $S_3$, the group of permutations acting on a set of size three, then the fixer of any element of the set is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ and is not a normal subgroup of $S_3$.