Minimal non-cyclic groups other than the Klein four groups [duplicate]

everyone. I am afraid that my question is too trivial. But here it is. The Klein four group is the first counterexample to the the statement: "If all proper subgroups of a group are cyclic, then the group is cyclic." I am looking for other examples, if any. Are there?

Thanks in advance.

Solution 1:

The example of the Klein group, $\mathbb{Z}_2 \times \mathbb{Z}_2$, generalizes naturally to $\mathbb{Z}_p \times \mathbb{Z}_p$ for any prime $p$. Other finite examples include any nonabelian group of order $pq$, where $p$ and $q$ are primes (such a group exists whenever $p-1$ is divisible by $q$).

But there are even infinite examples. For all sufficiently large primes $p$, there exist infinite groups all of whose proper nontrivial subgroups are cyclic of order $p$. These are the Tarski monsters.

Solution 2:

$\Sigma_3$ (the symmetric group on three elements) is the next smallest example.

Solution 3:

Another collection of examples, not yet mentioned, are the Prüfer groups (also known as quasicyclic groups); it is sometimes denoted $\mathbb{Z}_{p^{\infty}}$. They are infinite, but every proper subgroup is finite (and every nontrivial quotient is isomorphic to the original group). Here are three descriptions:

1. For a fixed prime $p$, let $\mathbb{Z}_{p^{\infty}}$ be the group of all $p^k$-th complex roots of unity for all $k\geq 0$, with the group operation being multiplication. It is not hard to show that every proper subgroup is generated by a $p^n$th primitive complex root of unity for some $n$, hence is cyclic.

2. As an alternative description, consider subgroup of the additive group $\mathbb{Q}/\mathbb{Z}$ that consists of all classes represented by a fraction whose denominator is a power of $p$.

3. A final description: consider the collection of groups $\{\mathbb{Z}/p^n\mathbb{Z}\}_{n\in\mathbb{N}}$, with injections $i_n\colon\mathbb{Z}/p^n\mathbb{Z}\to\mathbb{Z}/p^{n+1}\mathbb{Z}$ given by $i_n(a+p^n\mathbb{Z}) = pa+p^{n+1}\mathbb{Z}$. Then $\mathbb{Z}_{p^{\infty}}$ is the direct limit $\displaystyle\lim_{\longrightarrow}\mathbb{Z}/p^n\mathbb{Z}$ of this direct system.

The additive group $\mathbb{Q}$ is almost such a group, in that every finitely generated subgroup is cyclic, but of course it contains noncyclic proper subgroups (e.g., the pre-image of the Prüfer group for some $p$).