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I'm trying to find a proof of this:

The group $\langle\mathbb{Z}_n,\oplus\rangle$ is cyclic for every $n$, where $1$ is a generator. The generators of the group $\langle\mathbb{Z}_n,\oplus\rangle$ are all $g \in \mathbb{Z}_n $ for which $\gcd(g,n)=1$, as the reader can prove as an exercise.

It is perfectly clear that $1$ generates all $\mathbb{Z}_n$, but I can't get myself to understand the second part or find a way to prove it. Thanks.

Zev Chonoles
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f1sh3r0
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1 Answers1

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Hint:

Use Bézout's identity to prove that if $\,\gcd(g,n)=1$, $1$ can be obtained as a multiple of $g$ (in $\mathbf Z_n$).

Bernard
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  • I had a quick look at the bezout identity, but isn't there a simpler way? the text i quoted was from a first year undergrad discrete mathematics script, and the identity was never mentioned before in the text so i doubt this is the way the author imagined the proof, thank you though! – f1sh3r0 Jun 19 '15 at 12:56
  • Isn't Bézout's identity seen in high school? I'll think of a proof without it, but I won't be necessarily as simple… – Bernard Jun 19 '15 at 13:31
  • i didn't see it, but the referral to another question helped me understand the whole proof including Bézout's identity, so the question is closed - thanks again! – f1sh3r0 Jun 19 '15 at 13:35
  • Anyway, a direct proof would probably consist in proving a particular version of Bézout, i.e. reinventing the wheel. – Bernard Jun 19 '15 at 13:37
  • I believe Bézout's identity was covered in Chapter 4 of said script already. – Thomas Gassmann Nov 14 '21 at 18:45