Free Group Action

up:: Group Action

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

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 .

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 , then there’d be a non-identity element , which has to be the identity if the action is free.