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 .
This article is a stub. Help us out by expanding it.