Free Group Action

up:: Group Action

A group action of a Group $G$ over a set $X$ is said to be free if all non-identity elements of $G$ “push points away” (i.e. only the identity stabilizes points). That is,

$$\forall g\in G:\forall x\in X:g\cdot x=x⟹g=e_G$$

Equivalently, A group action is free iff all its stabilizers are trivial: only the identity element keeps elements intact, while all others “induce movement” upon $X$.

Properties

Free Group Actions induce a bijection between the group and a point’s orbit, since all group elements map to distinct points in the space. If it weren’t the case, i.e. there were $g\ne h\mid g\cdot x=h\cdot x=x$, then there’d be a non-identity element $(g^{-1}h):x\mapsto x$, which has to be the identity if the action is free.