Definition:Faithful Group Action/Definition 2

Definition

Let $G$ be a group with identity $e$.

Let $X$ be a set.

Let $\phi: G \times X \to X$ be a group action.

$\phi$ is faithful if and only if its permutation representation is injective.

Also known as

A faithful group action is also known as an effective group action.

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions
• 2003: David S. Dummit and Richard M. Foote: Abstract Algebra (3rd ed.) ... (next) Chapter $1$: Introduction to Groups: $\S 1.7$: Group Actions