Faithful group action

This article defines a group action property or a property of group actions: a property that can be evaluated for a group acting on a set.
View a complete list of group action properties|Get help on group action property lookup|Get exploration suggestions
VIEW RELATED: group action property implications | group action property non-implications | {{{context space}}} metaproperty satisfactions | group action metaproperty dissatisfactions | group action property satisfactions |group action property dissatisfactions

Definition

Definition in action terms

A group action of a group $G$ on a set $S$ is termed faithful or effective if for any non-identity elemnet $g\in G$, there is $s\in S$ such that $g.s\neq s$.

Definition in terms of homomorphisms

A group action of a group $G$ on a set $S$ is termed faithful or effective if the corresponding homomorphism from $G$ to $\operatorname {Sym} (S)$ is an injective homomorphism.