## If a Finite Group Acts on a Set Freely and Transitively, then the Numbers of Elements are the Same

## Problem 488

Let $G$ be a finite group and let $S$ be a non-empty set.

Suppose that $G$ acts on $S$ freely and transitively.

Prove that $|G|=|S|$. That is, the number of elements in $G$ and $S$ are the same.