Faithful Action

faithful action

If Group $G$ acts on a set $B$ and distinct element of $G$ induct distinct permutations of $B$ , the action is said to be faithful. A faithful Group Action is therefore one in which the associated permutation representation is injective.

$σ : G \to S_{X}$ is injective (here, it's all elements that get sent to a unique element, no duplicates allowed).

Some ways of interpreting this:

An action is faithful if $1_{G}$ is the only element that fixes every $x$ (ie: $Ker (σ) = {1_{G}}$ )
They are the distinct group elements that move the elements of $X$ in "distinct ways", namely $g_{1} \neq g_{2} \Rightarrow \exists x \in X (g_{1} \cdot x \neq g_{2} \cdot x)$ (this is the injective part)