Prove that $\Bbb{Z}/8\Bbb{Z}$ and $\Bbb{Z}/4\Bbb{Z}\times \Bbb{Z}/2\Bbb{Z}$ are not isomorphic.
Proof: We know that if $y\in\Bbb{Z}/8\Bbb{Z}$, then $\max_{y\in\Bbb{Z}/8\Bbb{Z}}\left\{ \text{ord}(y) \right\}=8$, but if $x\in\Bbb{Z}/4\Bbb{Z}\times\Bbb{Z}/2\Bbb{Z}$, then $\max_{x\in\Bbb{Z}/4\Bbb{Z}\times\Bbb{Z}/2\Bbb{Z}}\left\{ \text{ord}(x) \right\}=4$. Therefore $\Bbb{Z}/4\Bbb{Z}\times\Bbb{Z}/2\Bbb{Z}\not\cong\Bbb{Z}/8\Bbb{Z}$.
Is this a valid proof? Ifso can I do the same thing for $\Bbb{Z}/2\Bbb{Z}\times\Bbb{Z}/2\Bbb{Z}\times\Bbb{Z}/2\Bbb{Z}$, any tips/hints?