Why Of course such a group must be necessarily cyclic?

Question

Why the formula given in the answer here tells us that this group must be cyclic?

Here is the answer there:

In general, there is only one group of order $n$ iff gcd$(n,\varphi(n))=1$. Of course such a group must be necessarily cyclic. 561 satisfies the condition.

score 3 · Accepted Answer · answered May 26 '22 at 17:44

3

Because obviously there exists a cyclic group of order $n$ (namely $\{ 0, 1, 2, \ldots, n-1 \}$ with addition modulo $n$), so if there is a unique group of that order, it must be the cyclic one.

answered May 26 '22 at 17:44

Adayah

10,468

But I am trying to apply this result to a group of order 255 which is not prime. – Brain May 26 '22 at 17:46
Oh, are you using the fact that $\mathbb Z_n$ is a cyclic group for any $n$? – Brain May 26 '22 at 17:48
1

Yes. $\phantom{}$ – Adayah May 26 '22 at 17:48
Do you know any resource for a rigor proof for this fact mentioned in the solution? Is it a new fact? – Brain May 26 '22 at 17:50
I think it is given in a link in the comments – Brain May 26 '22 at 17:55
2

@Brain, no it is not "a new fact". It is a theorem of Dickson that is over 100 years old. See here. – Arturo Magidin May 26 '22 at 18:15

Why Of course such a group must be necessarily cyclic?

1 Answers1