Group of even order contains an element of order 2

Question

I am working on the following problem from group theory:

If $G$ is a group of order $2n$, show that the number of elements of $G$ of order $2$ is odd.

That is, for some integer $k$, there are $2k+1$ elements $a$ such that $a \in G,\;\; a*a = e$, where $e$ is the identity element of $G$.

Answer 1

Hint. Define an equivalence relation on $G$ by $x\sim y$ if and only if $x=y$ or $x=y^{-1}$. Then remember that an equivalence relation partitions a set.