Thread: Finite subgroups of Q/Z

How do I prove that every finite subgroup of Q/Z is cyclic?
Could someone help me?

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(b₁,...,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.