Is there any classification of finite group in which all proper subgroups are cyclic?
Would you please tell me a reference?
Is there any classification of finite group in which all proper subgroups are cyclic?
Would you please tell me a reference?
In the current wording, the answer to your question is exactly the class of all finite cyclic groups. However, in case you accidentally forgot the word "proper", there is the following classification of non-cyclic finite groups, such that all their proper subgroups are cyclic:
A finite group $G$ is a minimal noncyclic group if and only if $G$ is one of the following 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 $r ≡ 1 \pmod p$, $r^q ≡1 \pmod p$.
This theorem first appeared in "Non-abelian groups in which every subgroup is abelian" by G.A.Miller and H.G.Moreno (1903)