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Definition: Let n be a positive integer and consider the collection $R_n$ of all those integers m which satisfy $1 \leq m \leq n-1$, and for which the highest common factor of m and n is 1. We claim that multiplication modulo n makes $R_n$ into a group.

I've stuck on this problem for a long time. Currently I noticed that order of $R_p$ is ψ(p)=p-1, where ψ(p) is Euler's phi-function. Then we need to prove that there exists some element in $R_p$ such that order of this element is p-1. I listed some instances of p but haven't found a pattern to go further. Can anyone help?

turf
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1 Answers1

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The result you claim is a classical result called the primitive root theorem.

Generally, consider the ring $\mathbb{Z}/n\mathbb{Z}$. Then the group $R_n$ you define is precisely the multiplicative group of units for $\mathbb{Z}/n\mathbb{Z}$, which is commonly denoted $(\mathbb{Z}/n\mathbb{Z})^\times$. As you've noticed, this group has order $\varphi(n)$, where $\varphi$ is the Euler totient.

For the case where $n=p$ is a prime, the group turns out to be cyclic. In fact, the following result is known.

Theorem (Existence of Primitive Roots): Consider the group of units $(\mathbb{Z}/n\mathbb{Z})^\times$. Then this group is cyclic if and only if $n=2,4,p^k,2p^k$ for some integer $k$ and odd prime $p$.

There are multiple proofs for the existence of primitive roots for prime $p$, but none of them are particularly simple or short. Multiple proofs are contained here for example. Depending on how much group theory you know, the proof by Tunococ is probably the most accessible, but unfortunately also the longest.

In any case, you can find multiple proofs of varying flavours simply by searching "the primitive root theorem" on Google, for example. Particularly good are these notes written by Keith Conrad, which contains seven detailed proofs of desired result.

EuYu
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