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Given an action of a (discrete) group on a set , any set of the form for a fixed is called an orbit of the action, or the -orbit through the point . The set is a disjoint union of its orbits.
The category of orbits of a group is the full subcategory of the category of sets with an action of .
Since any orbit of is isomorphic to the orbit for some group , the category of -orbits admits the following alternative description: its objects are subgroups of and morphisms are elements such that .
In particular, the group of automorphisms of a -orbit is , where is the normalizer of in .
If is a topological group, a topological space and the action continuous, then one can distinguish closed orbits from those which are not. Even when one starts with Hausdorff, the space of orbits is typically non-Hausdorff. (This problem is one of the motivations of the noncommutative geometry of Connes’ school.)
If the original space is paracompact Hausdorff, then every orbit as a topological -space is isomorphic to , where is the stabilizer subgroup of .
Textbook accounts:
Last revised on April 18, 2021 at 16:22:18. See the history of this page for a list of all contributions to it.