nLab G-set

Idea

Given a topological group $G$, a (continuous) $G$-set is a set $X$ equipped with a continuous group action $\mu: G \times X \to X$, where $X$ is given the discrete topology.

In the case where $G$ is a discrete group, the continuity requirement is void, and this is just a permutation representation of the discrete group $G$.

Note that since $X$ must be given the discrete topology, this behaves rather unlike topological G-spaces. In particular, a topological group does not act continuously on itself, in general. Thus this notion is not too useful when $G$ is a “usual” topology group like $SU(2)$. Instead, the topology on the group acts as a filter of subgroups (where the filter contains the open subgroups), and each element of a continuous $G$-set is required to have a “large” stabilizer.

The $G$-sets form a category, where the morphisms are the $G$-invariant maps. See category of G sets.

Properties

Relation to $G$-orbits

Remark

($G$-sets are the free coproduct completion of $G$-orbits)
Let $G \,\in\, Grp(Set)$ be a discrete group. Since every G-set $X$ decomposes as a disjoint union of transitive actions, namely of orbits of elements of $X$, the defining inclusion of the orbit category into $G Set$ exhibits the latter as the free coproduct completion of the orbit category (see also this Prop.).

For topological groups

Proposition

Let $G$ be a topological group, and $X$ be a set with a $G$ action $\mu: G \times X \to X$. Then the action is continuous if and only if the stabilizer of each element is open.

Proof

Suppose $\mu$ is continuous. Since $X$ has the discrete topology, $\{x\}$ is an open subset of $X$. So $\mu^{-1}(\{x\})$ is open. So we know the stabilizer

$$I_x = \{g \in G: g \cdot x = x\} = \{g \in G: (g, x) \in \mu^{-1}(\{x\})\}$$

is open.

Conversely, suppose each such set is open. Given any (necessarily open) subset $A \subseteq X$, its inverse image is

$$\mu^{-1}(A) = \bigcup_{a \in A} \mu^{-1}(\{a\}).$$

So it suffices to show that each $\mu^{-1}(\{a\})$ is open. We have

$$\mu^{-1}(\{a\}) = \bigcup_{x \in X}\{g \in G: g \cdot x = a\} \times \{x\}.$$

Thus we only have to show that for each $a, x \in X$, the set $\{g \in G: g \cdot x = a\}$ is open. If there is no such $g$, then this is empty, hence open. Otherwise, let $g_0$ be such that $g_0 \cdot x= a$. Then we have

$$\{g \in G: g \cdot x = a\} = g_0 \cdot I_x.$$

Since $g_0$ is a homeomorphism, and $I_x$ is open, this is open. So done.

Examples

Discrete groups

In the following examples, all groups are discrete.

• A $\mathbb{Z}_2$-set is a set equipped with an involution.

• Any permutation $\pi : X \to X$ gives $X$ the structure of a $\mathbb{Z}$-set, with the action of $\mathbb{Z}$ on $X$ defined by iterated composition of $\pi$ or $\pi^{-1}$.

• $G$ is itself a $G$-set via the (left or right) regular representation.

• A normal subgroup $N \lhd G$ defines a $G$-set by the action of conjugation.

• For $G$ a finite group then Mackey functors on finite $G$-sets are equivalent to genuine G-spectra.

Continuous groups

• The group $\Sigma_N$ of permutations of the natural numbers can be given the topology generated by the stabilizers of finite subsets of $N$. This acts continuously on $N$. This is used in the construction of the basic Fraenkel model. See also nominal sets.

Related concepts

• topological G-space, simplicial group action

• category of G-sets

• class equation

• Galois theory

• action: a $G$-set is a set with an action of the given group, $G$.

• Borel model structure

• Burnside ring, Burnside category

• Burnside marks

• table of marks

References

An early account (where the term “representation group” is used to refer to a finite set equipped with a permutation action):

• William Burnside, On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups, Proceedings of the American Mathematical Society 1901 (doi:10.1112/plms/s1-34.1.159)

For a more modern account see

• Tammo tom Dieck, chapter 1 of Transformation Groups and Representation Theory, Springer 1979

Basic exposition of $G$-sets as a Grothendieck topos:

• Jaap van Oosten, (eg. p. 32 in:) Topos Theory (2018) [pdf, pdf]