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NonCommAlg our very own Prufer group: fix a prime number $\displaystyle p$ and define $\displaystyle G=\left \{\frac{m}{p^n} + \mathbb{Z}: \ \ m, n \in \mathbb{Z}, \ n \geq 0 \right \} \subset \mathbb{Q}/\mathbb{Z}.$ every proper subgroup of $\displaystyle G$ is finite and cyclic. in fact $\displaystyle H_n=<1/p^n + \mathbb{Z}>,$ where the
integer $\displaystyle n \geq 0,$ is fixed, are exactly all proper subgroups of $\displaystyle G.$ this is easy to prove and i'll leave it for you. a standard notation for $\displaystyle G$ is $\displaystyle \mathbb{Z}(p^{\infty}).$