Homomorphisms from p-adic units to the multiplicative cyclic groups

Question

What are the homomorphisms from $\mathbb{Z}_p^{\times} \to (\mathbb{Z}/n\mathbb{Z})^{\times}$, where $\mathbb{Z}_p$ denotes the p-adic integers?

Answer 1

From your answer to my comment, I see that you are not aware of the algebraic structure of $\Bbb Z_p^\times$. This is important, the key to your problem.

What happens is that $\Bbb Z_p^\times$ has a very small torsion subgroup, the roots of unity in $\Bbb Z_p$. That’s $\{\pm1\}$ in the case that $p=2$, and otherwise the $(p-1)$-th roots of unity, $p-1$ in number. In all cases, this group is cyclic.

And there’s an infinite, compact subgroup of $\Bbb Z_p^\times$, “pro-cyclic, pro-$p$”, which just means that it’s topologically generated by a single element, and isomorphic to the additive group $\Bbb Z_p$. In case $p=2$, this group may be conveniently generated by $5$, and in the general case, by $1+p$. To be more specific, this pro-cyclic subgroup is $1+4\Bbb Z_2$ or $1+p\Bbb Z_p$, depending on the prime. Most important fact about the infinite part is that its only quotients are cyclic $p$-groups.

I think that with this information, you can answer your question completely.

(If you’re wondering about the isomorphism between $1+4\Bbb Z_2$, resp. $1+p\Bbb Z_p$ and the additive group of $p$-adic integers, it’s given by way of the logarithm, defined by means of the series you know from Calculus for $\log(1+x)$. The image of the log will be the additive group $4\Bbb Z_2$, resp $p\Bbb Z_p$, both isomorphic to the full group of $p$-adic integers, as I’m sure you know.)