Let G be a finite group such that every element in G which isn't the identity has order of 2. Show that $|G| = 2^{n}$ for some $n \in \mathbb{N}$.
I know that G is necessarily going to be abelian. But what is a good method to prove the order of the group?
A good method is Cauchy's Theorem. This is slightly more high-tech than the approach showing that $G$ is abelian et cetera, but it's a much more adaptable result. For example, what about finite groups all of whose elements have order a power of $2$?