nLab
free action

Contents

Contents

Definition

Definition

A group action

()():G×XX (-)\cdot(-) \;\colon\; G \times X \to X

of a group GG on a set XX is called free if for every xXx \in X, the equation gx=xg \cdot x = x implies g=e Gg = 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,yXx,y \in X, there is at most one group element gGg \in G such that gx=yg \cdot x = y.

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

G×X (pr 2,) X×X (g,x) (x,gx). \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 GG acts freely on itself by multiplication :G×GG\cdot : G \times G \to G, which is called the (left) regular representation of GG.

  • An action of /2\mathbb{Z}/2\mathbb{Z} on a set XX corresponds to an arbitrary involution i:XXi : X \to X, but the action is free just in case ii is a fixed point-free involution.

  • For any set XX equipped with a transitive action *:G×XX* : G \times X \to X, the group Aut G(X)Aut_G(X) of GG-equivariant automorphisms of XX (i.e., bijections ϕ:XX\phi : X \to X commuting with the action of GG) acts freely on XX. In particular, suppose ϕAut G(X)\phi \in Aut_G(X) is such that ϕ(x)=x\phi(x) = x for some xXx\in X, and let yXy\in X be arbitrary. By the assumption that GG acts transitively, there is a gGg \in G such that y=g*xy = g*x. But then GG-equivariance implies that ϕ(y)=ϕ(g*x)=g*ϕ(x)=g*x=y\phi(y) = \phi(g*x) = g*\phi(x) = g*x = y. Since this holds for all yYy\in Y, ϕ\phi must be equal to the identity ϕ=id X\phi = id_X, and therefore Aut G(X)Aut_G(X) acts freely on XX.

  • A combinatorial species F:SetF : \mathbb{P} \to Set is said to be flat if all of the actions S n×F(n)F(n)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.