nLab
effective group action

this entry is about the concept in group theory; for the concept in quantumfield theory see at effective action functional; for disambiguation see effective action

Idea

A group action is effective if no group element other than the neutral element acts trivially on all elements of the space.

Definition

A group action of a group (group object) $G$ on a set (object) $X$ is effective if $\underset{x \in X}{\forall} g x = x$ implies that $g = e$ is the neutral element.

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

Related concepts

• transitive action, free action, semi-free action, regular action,

• proper action, properly discontinuous action

• effective orbifold