Problem 3

A group action is said to be free if $g\cdot x=x$ means that $g$ is the identity (i.e., only the identity fixes any $x$).

Let $G$ be a group and $S$ be a non-empty $G$-set. Suppose $G$ acts on $S$ freely and transitively. Prove that $G$ and $S$ have the same number of elements.