Discrete case

Given an action G×XXG\times X\to X of a (discrete) group GG on a set XX, any set of the form Gx={gx|gG}G x = \{g x|g\in G\} for a fixed xXx\in X is called an orbit of the action, or the GG-orbit through the point xx. The set XX is a disjoint union of its orbits.

Category of orbits

The category of orbits of a group GG is the full subcategory of the category of sets with an action of GG.

Since any orbit of GG is isomorphic to the orbit G/HG/H for some group HH, the category of GG-orbits admits the following alternative description: its objects are subgroups HH of GG and morphisms H 1H 2H_1\to H_2 are elements [g]G/H 2[g]\in G/H_2 such that H 1gH 2g 1H_1\subset g H_2g^{-1}.

In particular, the group of automorphisms of a GG-orbit G/HG/H is N G(H)/HN_G(H)/H, where N G(H)N_G(H) is the normalizer of HH in GG.

Topological case

If GG is a topological group, XX a topological space and the action continuous, then one can distinguish closed orbits from those which are not. Even when one starts with G,XG,X 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 GxG x as a topological GG-space is isomorphic to G/HG/H, where HH is the stabilizer subgroup of xx.



Textbook accounts:

Last revised on April 18, 2021 at 12:22:18. See the history of this page for a list of all contributions to it.