Coprime automorphism group implies cyclic with order a cyclicity-forcing number

Question

How do you prove Step 2 in Proof Outline, here?

Answer 1

If $P$ is the ($G$ is abelian by step $1$) Sylow $p$-subgroup of $G$ and $|P| \geq p^2$, then $G$ has a subgroup isomorphic to $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$ or isomorphic to $\mathbb{Z}/p^2\mathbb{Z}$ (these are the only non-isomorphic groups or order $p^2$). In the first case $Aut(G)$ has a subgroup isomorphic to $GL(2,p)$ and in the latter case to $\mathbb{Z}/p(p-1)\mathbb{Z}$. Since $|GL(2,p)|=(p^2-1)(p^2-p)$ in both cases this violates $gcd(|G|,|Aut(G)|)=1$ by a factor $p$.

Note: observe that this cyclicity forcing is equivalent to $gcd(\varphi(n),n)=1$, where $|G|=n$ and $\varphi$ Euler's totient function.