Is my argument correct, if not what is wrong with it. If the group $G$ is not cyclic, this means that there is not element of order 3. So if $G=\{e, a, b\}$ then $|a|=2$ and $|b|=2$, a contradiction because by Lagrange's Theorem element order divide group order but 2 does not divide 3. Hence $G$ is cyclic.
I just want to show if this is correct or not and why, I do not need a proof.