Suppose we have a finite abelian group $G$ and $x = \sum\limits_{g \in G}g$ .
Prove that $x \not= 0$ $<=>$ $Sylow-2$ of G is cyclic .
$<=$ was easy enough, because when the $sylow$ is cyclic it has a unique element of order 2. there are no more elements of order 2 outside the $sylow$ group, so the sum equlas that element. However, I can't seem to figure the other side. I think I need to show that there is more than 1 element of order 2 and that, those elements cancel each other in the sum, but haven't figured out how.