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.
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