# Difference between revisions of "Stabilizer"

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

Specifically, let <math>M</math> be a monoid operating on a set <math>S</math>, and let <math>A</math> be a subset of <math>S</math>. The ''stabilizer'' of <math>A</math>, sometimes denoted <math>\text{stab}(A)</math>, is the set of elements of <math>a</math> of <math>M</math> for which <math>a(A) \subseteq A</math>; the ''strict stabilizer''' is the set of <math>a \in M</math> for which <math>a(A)=A</math>. In other words, the stabilizer of <math>A</math> is the [[transporter]] of <math>A</math> to itself. | Specifically, let <math>M</math> be a monoid operating on a set <math>S</math>, and let <math>A</math> be a subset of <math>S</math>. The ''stabilizer'' of <math>A</math>, sometimes denoted <math>\text{stab}(A)</math>, is the set of elements of <math>a</math> of <math>M</math> for which <math>a(A) \subseteq A</math>; the ''strict stabilizer''' is the set of <math>a \in M</math> for which <math>a(A)=A</math>. In other words, the stabilizer of <math>A</math> is the [[transporter]] of <math>A</math> to itself. |

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

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

Specifically, let be a monoid operating on a set , and let be a subset of . The *stabilizer* of , sometimes denoted , is the set of elements of of for which ; the *strict stabilizer'* is the set of for which . In other words, the stabilizer of is the transporter of to itself.

By abuse of language, for an element , the stabilizer of is called the stabilizer of .

The stabilizer of any set is evidently a sub-monoid of , as is the strict stabilizer. Also, if is an invertible element of and a member of the strict stabilizer of , then is also an element of the strict stabilizer of , for the restriction of the function to is a bijection from to itself.

It follows that if is a group , then the strict stabilizer of is a subgroup of , since every element of is a bijection on , but the stabilizer need not be. For example, let , with , and let . Then the stabilizer of is the set of nonnegative integers, which is evidently not a group. On the other hand, the strict stabilizer of is the set , the trivial group. On the other hand, if is *finite*, then the strict stabilizer and the stabilizer are one and the same, since is bijective, for all .

**Proposition.** Let be a group acting on a set . Then for all and all , .

*Proof.* Note that for any , It follows that
By simultaneously replacing with and with , we have
whence the desired result.

In other words, the stabilizer of is the image of the stabilizer of under the inner automorphism .

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