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.