[[Group action]]
# Effective group action

A group action $\varphi : G \times M \to M$ is called **effective** or **faithful** iff the induced homomorphism $\Phi : G \to \mathrm{Aut}(M)$ is a [[group monomorphism]]. #m/def/group 
Equivalent conditions include

1. $gm = m$ for all $m \in M$ iff $g = e$

The terminology refers to the fact that if $G$ acts effectively then $G$ really does represent some group of symmetries of $M$ without redundancy.

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