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The primitive root theorem states that $U(n)$ is a cyclic group if and only if $n$ is $1, 2, 4, p^{k}, 2p^{k}$ where $p$ is an odd prime and $k$ is an integer greater than or equal to 1. Can someone please prove this for me? I was trying to prove this using the definition of group order, but that did not work.

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    $\begingroup$ The proof is not just a paragraph long. Fortunately, the full proof will be in many textbooks on elementary number theory (for example, the book of Niven/Zuckerman/Montgomery, Section 2.8). It's probably better to seek out that proof than to ask someone to copy pages of math down. $\endgroup$ – Greg Martin Oct 4 '16 at 4:04
  • $\begingroup$ Unfortunately I don't own the textbook, but a qualitative/intuitive proof would be okay with me. $\endgroup$ – PiccolMan Oct 4 '16 at 15:04
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Note that the relevant number theory term is "primitive root", which is a generator of the cyclic group $U(n)$ when that group is indeed cyclic. The general outline of establishing which moduli have primitive roots goes like this.

  • $1,2,4$ have primitive roots: do by hand.
  • Odd primes $p$ have primitive roots: follows from unique factorization of polynomials over $\Bbb F_p$ (shows that low orders of elements can't be too prevalent).
  • Powers of odd primes $p$: follows from "lifting" polynomial solutions (Hensel's lemma)
  • Twice a power of an odd prime: easy to show that $U(2p^k)$ is naturally isomorphic to $U(p^k)$ when $p$ is odd
  • $2^k$ with $k\ge3$: show that $\{\pm5^1,\pm5^2,\dots,\pm5^{2^{k-2}}\}$ form a reduced residue system, and in particular that every element has order at most $2^{k-2}$.
  • All other numbers can be factored as $jk$ where $(j,k)=1$ and $j,k>2$; in this case, all elements have order at most $\frac12\phi(jk) = \phi(j)\frac{\phi(k)}2 = \frac{\phi(j)}2\phi(k)$.
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