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group theory

Contents

Definition

Discrete case

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

Category of orbits

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

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

In particular, the group of automorphisms of a $G$-orbit $G/H$ is $N_G(H)/H$, where $N_G(H)$ is the normalizer of $H$ in $G$.

Topological case

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

References

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.