What is difference between free and faithful group action?

Question

I searched Wikipedia for definitions of free and faithful actions. As I understand them, the two concepts are the same thing!

If they are one concept, what is the point of introducing both or even of naming them in distinct ways?

Answer 1

Free mean that if there is $x \in X$ and $g,h$ with $gx = hx$ then $g = h$. Faithful means that the morphism $G \to Sym(X)$ induced by the action is injective, i.e for all $g\ne h$ there is a $x \in X$ with $gx \neq hx$.

Of course, being free is stronger. It's not equivalent since the action of $\text{SO}(2)$ on $\Bbb R^2$ this is not free since there is a fixed point but it's faithful (take $x = (1,0)$ works for all $g,h \in \text{SO}(2)$).