How do I prove that every finite subgroup of Q/Z is cyclic?

Could someone help me?

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- Nov 11th 2012, 09:53 PMRitaFinite subgroups of Q/Z
How do I prove that every finite subgroup of Q/Z is cyclic?

Could someone help me? - Nov 12th 2012, 12:16 AMDevenoRe: Finite subgroups of Q/Z
let G be a finite subgroup of Q/Z. since G is finite, it is finitely generated by some finite set of cosets {q

_{i}+Z: i = 1,2,...,n}.

suppose we pick each q_{i}in (0,1) and write q_{i}= a_{i}/b_{i}, with gcd(a_{i},b_{i}) = 1, for each i.

let m = lcm(b_{1},...,b_{n}).

we then have that G is contained in the cyclic group <(1/m)+Z>, and thus, being a subgroup of a cyclic group, is itself cyclic.