# Difference between revisions of "Fixer"

(new page) |
m (Link to group action article) |
||

Line 1: | Line 1: | ||

− | A '''fixer''' is part of a [[monoid]] (or [[group]]) acting on a [[set]]. | + | A '''fixer''' is part of a [[monoid]] (or [[group]]) [[group action|acting]] on a [[set]]. |

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

## Latest revision as of 17:47, 9 September 2008

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. ! Extra }, or forgotten $.), 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 .