Example Of a Proper noncyclic Subgroup of rationals

Question

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?

Answer 1

The dyadic rationals

$$\left\{\frac{a}{2^n} : a, n \in \mathbb{Z}\right\}$$

form a subgroup of $(\mathbb{Q}, +)$, but they aren't generated by a single element (if it were generated by some $r$, what happens to $r/2$?).

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This can be done with $2$ replaced by other things, e.g. other primes.

Answer 2

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 $\,q=1/2).$