TITLE: Other method(s) to prove "a group cannot have exactly two elements of order $2$"
QUESTION [5 upvotes]: If $a,b$ are elements of a group having order $2$ then, if $a,b$ commute, $ab(\ne a , \ne b)$ has order $2$, and if $a,b$ do not commute, then $aba^{-1}(\ne a , \ne b, \ne e)$ has order $2$. Using this we can deduce "a group cannot have exactly two elements of order $2$".

Is there any other method to prove this statement?

REPLY [3 votes]: You can make McKay's proof work in the general (not necessarily finite) case by referencing Dietzmann's Lemma.  Namely, if any group has a finite number of order $p$ elements, the number of such elements is congruent to $-1\pmod{p}$.