As part of my study of Abstract Algebra, I’m trying to prove that $ U_{p} $ is cyclic for $ p $ a prime number. It’s a classical result, but I’m trying to prove it following four steps stated as problems in a book. I was able to handle Problems $ 1 $ and $ 2 $, but $ 3 $ is giving me trouble.
Problem $ 3 $. Let $ G $ be a finite abelian group of order $ n $ for which the number of solutions of the equation $ x^{m} = e_{G} $ is at most $ m $ for any $ m $ dividing $ n $. Prove that $ G $ must be cyclic.
Hint: Let $ \psi(m) $ be the number of elements of $ G $ of order $ m $. Show that $ \psi(m) \leq \phi(m) $ and use Problem $ 2 $. ($ \phi $ is Euler’s totient function.)
The other two problems are:
Problem $ 1 $. In a cyclic group of order $ n $, show that for each positive integer $ m $ that divides $ n $, there are $ \phi(m) $ elements of order $ m $.
Problem $ 2 $. Using Problem $ 1 $, show that $ \displaystyle n = \sum_{m|n} \phi(m) $.
I sincerely have no clue on how to solve, or merely attack, Problem $ 3 $, so I came here looking for a better hint. I’m supposed to use only basic group-theoretic results up to Lagrange’s Theorem.
Note: It’s not homework, just personal study. The book is I. N. Herstein’s Abstract Algebra.