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stabilizer (Definition)

Let $ G$ be a group, $ X$ a set, and $ \cdot: G \times X \longrightarrow X$ a group action. For any subset $ S$ of $ X$, the stabilizer of $ S$, denoted $ \operatorname{Stab}(S)$, is the subgroup

$\displaystyle \operatorname{Stab}(S) := \{g \in G \mid g\cdot s \in S$   for all $\displaystyle \ s \in S\}. $
The stabilizer of a single point $ x$ in $ X$ is often denoted $ G_x$.



"stabilizer" is owned by djao.
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Other names:  isotropy subgroup

Attachments:
conjugate stabilizer subgroups (Derivation) by Thomas Heye
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Cross-references: point, subgroup, subset, group action, group
There are 31 references to this entry.

This is version 3 of stabilizer, born on 2002-01-21, modified 2003-03-23.
Object id is 1518, canonical name is Stabilizer.
Accessed 6131 times total.

Classification:
AMS MSC20M30 (Group theory and generalizations :: Semigroups :: Representation of semigroups; actions of semigroups on sets)
 16W22 (Associative rings and algebras :: Rings and algebras with additional structure :: Actions of groups and semigroups; invariant theory)

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