# Math Help - Finite subgroups of Q/Z

1. ## Finite subgroups of Q/Z

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

2. ## Re: 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 {qi+Z: i = 1,2,...,n}.

suppose we pick each qi in (0,1) and write qi = ai/bi, with gcd(ai,bi) = 1, for each i.

let m = lcm(b1,...,bn).

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.