Structure of the multiplicative group of a finite field

Question

Let $p$ be prime and $n\in\mathbb{N}$. Is it true that $\left(\mathbb{F}_{p^n}\right)^\times\simeq \mathbb{Z}_{p^n-1}$, when looking at $\mathbb{Z}_{p^n-1}$ additively?Furthermore, assume that $p^n-1$ is prime. Does in this case the above isomorphism still hold, when looking at $\mathbb{Z}_{p^n-1}$ multiplicatively?

Answer 1

1) Yes. If $F$ is a field and $G$ a finite subgroup of $F^\times$, then $G$ is cyclic. This follows because all elements of $G$ must be roots of the polynomial $X^{|G|}-1$ and that polynomial has at most (hence exactly) $|G|$ roots.

2) No. Multiplicatively, $\Bbb Z_{p^n-1}$ is not even a group.