A group action is free iff all its stabilizers are trivial

up:: Free Group Action

$(⟹)$

Let there be a Free Group Action of $G$ over $X$. Let $x\in X$. Then the Stabilizer of $x$ is

$$Sta{b}_x=\{g\in G\mid g\cdot x=x\}$$

However, by the definition of a free group action, we have that $g\cdot x=x⟹g=e_G$. Thus, $Sta{b}_x=\{e_G\}$.

Thus, the stabilizers of all $x\in X$ are trivial.

$(⟸)$

Let $x\in X$, for which we assume $Sta{b}_x=\{e_G\}$.

Thus, this means that $\forall g\in G,g\cdot x=x⟹g=e_G$, which means that the group action is free.