Effective group action

A group action $𝜑 : 𝐺 \times 𝑀 \to 𝑀$ is called effective or faithful iff the induced homomorphism $Φ : 𝐺 \to A u t (𝑀)$ is a group monomorphism. #m/def/group Equivalent conditions include

$𝑔 𝑚 = 𝑚$ 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.

#state/tidy | #lang/en | #SemBr