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 $\mathfrak{S}_A}$ , 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 $S_3$ .

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Fixer

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