Here are some excerpts from my lecture note:
- An action from a group $G$ to a set $X$ is a homomorphism $\alpha: G \to \text{Sym}(X)$, where $\text{Sym}(X)$ is the group of bijections $X \to X$.
- Equivalently, an action $\alpha$ of $G$ on $X$ can be viewed as a map $G \times X \to X$, which we usually write as $(g,x) \mapsto g \cdot x := \alpha_g(x) \in X$.
- An action $\alpha$ is called faithful if it is injective.
- The action of left translation of a group $G$ on itself is defined as $L_g(h) = gh$, for some fixed $g \in G$. This action is faithful.
Am I correct to assume that the statement that $L_g$ is faithful is vacuously true, since we are restricting the domain of the action to just one element $g$? Or am I missing something?
It's $L$ that is faithful, not $L_g$.
That each $L_g$ is a bijection is indeed obvious. To show $L$ is faithful you need to show that $L_g \ne L_h$ when $g \ne h$.
That is not hard.