TITLE: Example Of a Proper noncyclic Subgroup of rationals
QUESTION [8 upvotes]: We know the set of rational numbers forms a group under addition. My question is :does there exist a proper subgroup of rationals which is not cyclic? If yes, how can we construct it?

REPLY [5 votes]: Hint $\ $ A cyclic subgroup has form $\,q\,\Bbb Z\,$ so has bounded denominators. So to get a noncyclic subgroup of $\,\Bbb Q\,$  it suffices to choose any subgroup having unbounded denominators. For example, any subring of $\,\Bbb Q\,$ (except $\,\Bbb Z)\,$ works since it contains a proper fraction $\,q\not\in\Bbb Z\,$ whose powers have unbounded denominator. A simple choice is the subring generated by $q,\,$ i.e. $\,\Bbb Z[q] = $ all polynomials in $\,q\,$ with integer coefficients (the dyadics in T.Bongers answer is the special case $\,q=1/2).$