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 $F_2$.
