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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 $$ \operatorname{Stab}(S) := \{g \in G \mid g\cdot s \in S \text{for all }\ s \in S\}. $$ The stabilizer of a single point $x$ in $X$ is often denoted $G_x$
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"stabilizer" is owned by djao.
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| Other names: |
isotropy subgroup |
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Cross-references: point, subgroup, subset, group action, group
There are 33 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 7700 times total.
Classification:
| AMS MSC: | 20M30 (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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Pending Errata and Addenda
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