Faithful group action

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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 \operatorname{Sym}(S) is an injective homomorphism.