Is it possible to prove that $U(p)$, for prime $p$, is cyclic using only group theory? If not, why not?

Question

The unitary group modulo $p$ is also written $(Z/pZ)^*$ and includes the integers $U(p) = \{ 1, 2, …, p - 1 \}$ and is a group under multiplication modulo $p$.

While this is not the exact problem I'm working on, it is most definitely an important piece of a homework problem, so please, hints are prefered.

I have found several proofs of the existence of a primitive root, which show that $U(p)$ is cyclic. However, they all rely on polynomials and field theory. I'm looking for a proof that does not require these subjects. Is one possible? If it's not possible, could someone show why it's not possible?

Answer 1

Consider this lemma, which is pure group theory (*):

Let $G$ be a group of order $m$. If, for every divisor $d$ of $m$, there are no more than $d$ elements of $G$ satisfying $x^d=1$, then $G$ is cyclic.

The result follows directly from this lemma because $(\Bbb Z/p\Bbb Z)^*$ is a field and a polynomial of degree $d$ over a field has at most $d$ roots. That is the arithmetic part of the result.

(*) This lemma appears verbatim in chapter X of André Weil's Number theory for beginners, which is a wonderful book.