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1 [made Community Wiki]

A bit rubbish and easy, but amusing if you haven't seen it before.

Let $G$ be a finite group such that $g^2=e$ for all $g \in G$, i.e. every element (except the identity $e$) has order 2. Then $G$ has size $2^n$ for some $n$.

This is not too hard to prove directly; but it becomes totally obvious (once you've proved that $G$ is abelian) when you realise that $G$ is a finite-dimensional vector space over $F_2$.