# Faithful group action

From Groupprops

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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## Definition

### Definition in action terms

A group action of a group on a set is termed **faithful** or **effective** if for any non-identity elemnet , there is such that .

### Definition in terms of homomorphisms

A group action of a group on a set is termed **faithful** or **effective** if the corresponding homomorphism from to is an injective homomorphism.