If $G$ is Abelian group and $a,b \in G$ are distinct elements of order $2$, show that $ab$ has order $2$. Prove that $H=\{e,a,b,ab\}$ forms a subgroup of $G$ that is not cyclic.
I request help on how I can show that $ab$ has order $2$ as well as to show $H$ is not cyclic. I think there exist no $x \in G$ that generates $H$.
Thanks all.