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. | ||
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<cmath> \text{stab}(ax) \subseteq a\, \text{stab}(x) a^{-1} , </cmath> | <cmath> \text{stab}(ax) \subseteq a\, \text{stab}(x) a^{-1} , </cmath> | ||
whence the desired result. <math>\blacksquare</math> | whence the desired result. <math>\blacksquare</math> | ||
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+ | In other words, the stabilizer of <math>ax</math> is the image of the stabilizer of <math>x</math> under the [[inner automorphism]] <math>\text{Int}(a)</math>. | ||
{{stub}} | {{stub}} |
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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