free action



An action

ρ:G×XX \rho : G \times X \to X

of a group GG on a set XX is free if for every xXx \in X and every gGg\in G, the equation gx=xg x = x implies g=1 Gg=1_G. Equivalently, an action is free when for any pair of elements x,yXx,y \in X, there is at most one group element gGg \in G such that gx=yg x = y.

A free action that is also transitive is called regular.


  • 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.

  • A free action of /2\mathbb{Z}/2\mathbb{Z} on a set XX is the same thing as a fixed point free involution on XX.

  • 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.

Revised on October 12, 2016 11:58:11 by Noam Zeilberger (