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.
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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 $g.s\ne s$.

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 $\mathrm{S}\mathrm{y}\mathrm{m}(S)$ is an injective homomorphism.