TITLE: Number of elements of order $2$ in Abelian groups of order $2^{n}$
QUESTION [5 upvotes]: I'm self studying some group theory and one of the exercises I came across:
Question: Prove that an Abelian group of order $2^{n}, n \in \mathbb{N}$ must have an odd number of elements of order 2. 
I'm not sure how to approach this problem. Any hints would be appreciated. Prefer no complete answers but could use one for self checking. Thank you.

REPLY [2 votes]: Hint: define a relation on $G$ by
$$
a\sim b\quad\text{if and only if}\quad b=a\text{ or }b=a^{-1}
$$
Prove this is an equivalence relation and that the equivalence classes have one or two elements. Which ones have just one element?