For $p$ prime, the group $R_p$ is cyclic

Question

The group $R_p$ is the group of all integers $q$ with $1\leq q \leq p-1$, where the greatest common factor of $p$ and $q$ is 1.

I'm aware of this proof but I was thinking of an alternate route. What if I could define a mapping from $R_p$ to $\mathbb{Z_{p-1}}$ and show that it is an isomorphism. Then the "cyclicity" of $\mathbb{Z_{p-1}}$ carries over to $R_p$. How could I define such a mapping? Is it possible?

I assume $p$ is prime, right? $R_p$ has $p - 1$ elements and $\mathbb{Z}_{p - 2}$ has $p - 2$ elements, so no isomorphism between them exists. — Ethan Alwaise, Feb 23 '16 at 05:19
@JacopoStifani how is $\mathbb{Z}_{p-1}$ defined? In the standard definition, this is trivial — Stella Biderman, Feb 23 '16 at 05:25
Your link is to a proof that uses $\sum_{d|n}\phi (d)=n.$ Would you like to see an elementary proof that does not use it? — DanielWainfleet, Feb 23 '16 at 06:21
In number-theoretic language, we need to show that there is a primitive root mod $p$, and then the discrete logarithm does the job. — André Nicolas, Feb 23 '16 at 06:40

score 2 · Answer 1 · answered Feb 23 '16 at 06:48

Finding an isomorphism $R_p \to \mathbb Z_{p-1}$ is the same as finding a generator of $R_p$. If you are this point, you have already shown that $R_p$ is cyclic. There is no "totally trivial" proof of this fact. The proof will always be the same work as the proof that any finite subgroup of $K^*$ (for a field $K$) is cyclic.

Stella Biderman · Answer 2 · 2016-02-23T05:29:43.580

1

The "identity" map works. Map the integer $a$ to the coset $a+p\mathbb{Z}$, which is clearly an isomorphism.

edited Feb 23 '16 at 05:29

answered Feb 23 '16 at 05:19

Stella Biderman

31,155

Yes indeed, i meant $\mathbb{Z_{p-1}}$. – Jacopo Stifani Feb 23 '16 at 05:22
@JacopoStifani does this do it – Stella Biderman Feb 23 '16 at 05:42
But $\mathbb{Z_{p-1}}$ has order $p$ and $R_p$ has order $p-1$. How will this work? – Jacopo Stifani Feb 28 '16 at 04:17

For $p$ prime, the group $R_p$ is cyclic

2 Answers2