lifting generators of multiplicative groups of prime power

Question

I'm trying to understand a proof when the multiplicative group of integers modulo n is cyclic. It starts by supposing $g + p \mathbb{Z}$ is a generator in $(\frac{\mathbb{Z}}{p\mathbb{Z}})^{\times}$. Then $g + p \mathbb{Z}$ has order p-1.
until here, I understand the reasoning.

Then it continues: the order of $g + p^{e} \mathbb{Z}$ in $(\frac{\mathbb{Z}}{p^{e} \mathbb{Z}})^{\times}$ is a multiple of p-1. I tried this with a couple of examples but i didn't quite get the hang of it. Could someone elaborate how to connect these two rings?

Thanks in advance!

Answer 1

$\def\z{\Bbb Z}$Denote the order of $g+p^e\z$ in $(\z/p^e\z)$ as $m$.

Then $g^m\equiv1\pmod{p^e}$, which implies $g^m\equiv1\pmod p$.

Hence $m$ is a multiple of $p-1$, the order of $g$ in $\z/p\z$.

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Hope this helps.