nLab
free action

Contents

Definition

* \;\colon\; G \times X \to X

of a group $G$ on a set $X$ is called free if for every $x \in X$ , the equation $g \ast x = x$ implies $g= e_G$ (the neutral element).

Equivalently, an action is free when for any pair of elements $x,y \in X$ , there is at most one group element $g \in G$ such that $g * x = y$ .

Beware the similarity to and difference with effective action: a free action is effective, but an effective action need not be free.

A free action that is also transitive is called regular.

Any group $G$ acts freely on itself by multiplication $\cdot : G \times G \to G$ , which is called the (left) regular representation of $G$ .
An action of $\mathbb{Z}/2\mathbb{Z}$ on a set $X$ corresponds to an arbitrary involution $i : X \to X$ , but the action is free just in case $i$ is a fixed point-free involution.
For any set $X$ equipped with a transitive action $* : G \times X \to X$ , the group $Aut_G(X)$ of $G$ -equivariant automorphisms of $X$ (i.e., bijections $\phi : X \to X$ commuting with the action of $G$ ) acts freely on $X$ . In particular, suppose $\phi \in Aut_G(X)$ is such that $\phi(x) = x$ for some $x\in X$ , and let $y\in X$ be arbitrary. By the assumption that $G$ acts transitively, there is a $g \in G$ such that $y = g*x$ . But then $G$ -equivariance implies that $\phi(y) = \phi(g*x) = g*\phi(x) = g*x = y$ . Since this holds for all $y\in Y$ , $\phi$ must be equal to the identity $\phi = id_X$ , and therefore $Aut_G(X)$ acts freely on $X$ .
A combinatorial species $F : \mathbb{P} \to Set$ is said to be flat if all of the actions $S_n \times F(n) \to F(n)$ are free (see Combinatorial species and tree-like structures). For example, the species of linear orders is flat.

Last revised on November 23, 2020 at 12:14:40. See the history of this page for a list of all contributions to it.