I am trying some exercises on a course about algebraic number theory and I am slightly confused about the question.
First we are told to show that $U_n/U_{n+m}\cong\mathbb{Z}/p^m\mathbb{Z}$ where $U_i=1+p^i\mathbb{Z}_p$ for sufficiently large $n$.
The second part asks us to "use this to prove that if $p$ is odd, $(\mathbb{Z}/p^m\mathbb{Z})^*$ is cyclic. "
Edit: This is the actual question as it is stated.
Let $U_r=1+p^r\mathbb{Z}_p$. Show that for $r$ sufficiently large, $U_r/U_{r+m}\cong \mathbb{Z}/p\mathbb{Z}$. Use this to prove that if $p$ is odd then $(\mathbb{Z}/p\mathbb{Z})^*$ is cyclic.
My attempt: The first part I was able to show an isomorphism but only an isomorphism of groups.
As a set $U_n/U_{n+m}=\{1+ap^n+p^{n+m}\mathbb{Z}:a\in\{1,2,...,p^m\}\}$
Take $x=1+ap^n+p^{n+m}\mathbb{Z}_p$
$y=1+bp^n+p^{n+m}\mathbb{Z}_p$ Then, $xy=1+(a+b)p^n+p^{2n}(a+b)+p^{n+m}\mathbb{Z}_p$
So if we take $n\geqslant m$ , we get $xy=1+(a+b)p^n+p^{n+m}\mathbb{Z}_p$
And we have an isomorphism (of groups)
$U_n/U_{n+m}\to \mathbb{Z}/p^m\mathbb{Z}$ given by $1+ap^n+p^{n+m}\mathbb{Z}_p\mapsto \bar{a}$
Hence, I am slightly confused the second part as $U_n/U_{n+m}$ has no multiplicative structure so as to speak and thus I cannot the isomorphism in (i). How do I endow the multiplicative structure on $U_n/U_{n+m}$?
(I would also appreciate if someone could tell me if there is a way to do this more abstractly such as using the isomorphism theorem for a particular mapping. Also, no where in the question have I used anything p-adic. Can I just not replace $U_i$ with $1+p^i\mathbb{Z}$ and still get the same thing?)