existence of $a \neq e$ in group such that $a^2=e$ $G$ - group, $|G|=2k$, $k\in \Bbb N$
Does there exist $a\in G; a \neq e: a^2 =e$?
I think I should somehow use the fact, that there is odd number of elements of $G$ which are not $e$.

if $a^2 \ne e$, then $a \ne a^{-1}$.
there are $2k-1$ elements of $G$ which are not $e$, so these exist odd number of elements of $G$ which satisfy $a^2=e$ but $a\ne e$
