Cyclic group of order n

Question

Let G be a cyclic group of order n, and let r be an integer dividing n. Prove that G contains exactly one subgroup of order r

Probable duplicate of https://math.stackexchange.com/questions/410389/subgroups-of-a-cyclic-group-and-their-order — lhf, Oct 08 '19 at 16:52

score 0 · Accepted Answer · answered Oct 08 '19 at 16:58

A subgroup of a cyclic group is cyclic (You should be able to see that, but if you have problems with that, let me know). Then take any subgroup $H$ of order $r$ diving $n$ and take $g$ a generator of the whole group. Then set

$$ s=min\{k\in\mathbb Z| g^k\in H\} $$

Then you can prove $H=\left<g^s\right>$ (by contradiction for instance). And as $H$ has order $r$ we bust have that the order of $g^s$ is $r$ and you can conclude $s=n/r$. I leave you to fill the details, but that should do it.

Cyclic group of order n

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