# Which of the following group has a proper subgroup that is not cyclic?

Which of the following group has a proper subgroup that is not cyclic?

$1$. $\mathbb Z_{15} \times \mathbb Z_{17}$.

$2$. $S_3$

$3$. ($\mathbb Z$,+)

$4$. ($\mathbb Q$,+)

• the proper subgroups of $\mathbb Z_{15} \times \mathbb Z_{17}$ have possible orders $3,5,15,17,51,85$ & all groups of orders $3,5,15,17,51,85$ are cyclic.So,all proper subgroups of $\mathbb Z_{15} \times \mathbb Z_{17}$ are cyclic.
• Every proper subgroup of $S_3$ is cyclic.So,it is not the answer.
• ($\mathbb Z$,+) is a cyclic group generated by $1$.And every proper subgroup of a cyclic group is cyclic.So,it is not the answer.
• Any finitely generated subgroup of ($\mathbb Q$,+) is cyclic.So,it is not the answer.

The answer given in the answer key is $4$.

• Why did you restrict yourself to finitely generated subgroups of $\mathbb{Q}$? – carmichael561 May 27 '17 at 4:17
Hint. As regards 4. what about $\left\{\frac{m}{2^n} : m, n \in \mathbb{Z}\right\}$.