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 $4$.
Please help me knowing which point i'm missing.
Best Answer
Hint. As regards 4. what about $\left\{\frac{m}{2^n} : m, n \in \mathbb{Z}\right\}$.