Contents

group theory

# Contents

## Definition

###### Definition
$(-)\cdot(-) \;\colon\; G \times X \to X$

of a group $G$ on a set $X$ is called free if for every $x \in X$, the equation $g \cdot x = x$ implies $g = e_G$ (the neutral element), hence if only the action of the neutral element has fixed points.

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

This means equivalently that the action is free if and only if its shear map is a monomorphism:

$\array{ G \times X & \xhookrightarrow{\;(pr_2, \cdot)\;} & X \times X \\ (g, x) & \mapsto & \big( x, g \cdot x \big) \,. }$

In this form the definition makes sense for action objects internal to any ambient category (with finite products).

###### Remark

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

###### Remark

A free action that is also transitive is called regular.

## Examples

• 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 September 16, 2021 at 23:32:52. See the history of this page for a list of all contributions to it.