# Is there some sort of classification of all minimal non-cyclic groups?

Does there exist some sort of classification of all minimal non-cyclic groups (non-cyclic groups, such that all their proper subgroups are cyclic)

I know the following classes of such groups:

1) $$C_p × C_p$$, where $$p$$ is a prime

2) $$Q_8$$

3) $$\langle a,b | a^p = b^{q^m} = 1, b^{−1}ab = a^{r}\rangle$$, where $$p$$ and $$q$$ are distinct primes and $$r ≡ 1 \pmod p$$, $$r^q ≡1 \pmod p$$.

(These three classes completely cover the case, when our group is finite: Classification of finite minimal non-cyclic group)

4)$$C_{p^{\infty}}$$, where $$p$$ is a prime

5)$$(\{ \frac{n}{p^m}| m, n \in \mathbb{Z} \}, +)$$, where $$p$$ is a prime

(These two classes completely cover the case, when our group is infinite abelian: Does there exist an infinite non-abelian group such that all of its proper subgroups become cyclic?)

6)Infinite non-abelian groups, such that all their nontrivial proper subgroups are isomorphic to $$C_{p}$$ for a fixed prime $$p$$ (Tarski monster groups)

7)Infinite non-abelian groups, such that all their nontrivial proper subgroups are isomorphic to $$C_{\infty}$$ (Does there exist an infinite non-abelian group, such that all its nontrivial proper subgroups are isomorphic to $C_\infty$?).

However, I do not know, whether there exists anything, that does not fall into these classes. I only know, that if such groups exist, they have to be infinite non-abelian.

• – lhf May 3 at 14:34

For every sufficiently large odd number $$n$$ there is a 2-generator infinite group $$G(n)$$ such that every maximal (proper) subgroup of $$G(n)$$ is cyclic of order $$n$$.
Now, take a non-prime odd number $$n$$ and apply this theorem. You obtain a 2-generator group where every proper subgroup is cyclic, but orders of these cyclic subgroups are nonconstant (depend on the subgroup).