# Stabilizer

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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