Fixer
A fixer is part of a monoid (or group) acting on a set.
Specifically, let be a monoid acting on ; let be a subset of . The fixer of is the set of all fow which for all . If contains a single element , we sometimes call this the fixer or stabilizer of , by abuse of language.
Evidently, the fixer of is a submonoid of (and of the strict stabilizer of ). Also, if is an invertible element of the fixer of , then is evidently an element of the fixer of .
It follows that if is a group , then the fixer of is a subgroup of . In fact, it is a normal subgroup of the strict stabilizer of . Indeed, it is the kernel of the canonical homomorphism from the strict stabilizer of to $\mathfrak{S}_A}$ (Error compiling LaTeX. Unknown error_msg), the group of permutations on .
Note, however, that need not be a normal subgroup of . For example, if is , the group of permutations acting on a set of size three, then the fixer of any element of the set is isomorphic to and is not a normal subgroup of .