orbit (changes) in nLab

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Definition

Discrete case

Given an action G×X→XG\times X\to X of a (discrete) group GG on a set XX, any set of the form Gx={gx|g∈G}G x = \{g x|g\in G\} for a fixed x∈Xx\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 1→H 2H_1\to H_2 are elements [g]∈G/H 2[g]\in G/H_2 such that H 1⊂gH 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.

Examples

• An orbit of a cyclic subgroup of a permutation group is called a permutation cycle.

• orbit category, orbit type

• coadjoint orbit

• coinvariant

• coset space

• orbit method

• slice theorem

References

Textbook accounts:

• Glen Bredon, Sections I.3, I.4 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)