It is well known that $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$ is a cyclic group for a prime $p>2$ and $n\geq 1$. However, most of the proofs are a little complicated, and I want to find some neat proof of this.
It is also known that the unit group $\mathbb{Z}_{p}^{\times}$ of $p$-adic integers is isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{\times}\times (1+p\mathbb{Z}_{p})\simeq \mathbb{Z}/(p-1)\mathbb{Z}\times \mathbb{Z}_{p}$, where the latter isomorphism $(1+p\mathbb{Z}_{p}, \bullet)\simeq( \mathbb{Z}_{p}, +)$ is given by $\log$ map. I want to use this to prove that $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$ is cylic.
In fact, a sort of converse holds: assume that we knows $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}\simeq \mathbb{Z}/p^{n-1}(p-1)\mathbb{Z}$. Then we have $$ \mathbb{Z}_{p}^{\times} = \lim_{\longleftarrow}(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}\simeq (\mathbb{Z}/(p-1)\mathbb{Z})\times \lim_{\longleftarrow} \mathbb{Z}/p^{n-1}\mathbb{Z} \simeq \mathbb{Z}/(p-1)\mathbb{Z}\times \mathbb{Z}_{p}, $$ so we may reprove that $\mathbb{Z}_{p}^{\times}$ is isomorphic to $\mathbb{Z}/(p-1)\mathbb{Z} \times \mathbb{Z}_{p}$. Is there any nice way to reverse this argument so that we can prove $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times} \simeq\mathbb{Z}/p^{n-1}(p-1)\mathbb{Z}$? Thanks in advance.
