Is there any classification of finite group in which all proper subgroups are cyclic?

Would you please tell me a reference?

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    $\begingroup$ Do you mean proper subgroups? $\endgroup$ – Benjamin Steinberg Nov 2 '16 at 22:00
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    $\begingroup$ In addition to the cyclic groups themselves, you would be looking at the (finite) minimal non-cyclic groups. Geoff Robinson gives an answer at math.stackexchange.com/questions/1934131/… They are: $Q_8$, $C_p\times C_p$, and the unique group of order $pq^n$ with $p\equiv 1\pmod{q}$ with a normal Sylow $p$-subgroup and cyclic groups of order $pq^{n-1}$ and $q^n$. $\endgroup$ – Arturo Magidin Nov 2 '16 at 22:37
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    $\begingroup$ The finite groups those abelian subgroups are cyclic have been classified by Zassenhaus and Suzuki (these are exactly the finite groups with periodic integral cohomology). There are exactly 6 types. A list can be found in Adem, Milgram: Cohomology of finite Groups, Theorem 6.15. Then you can check which of them have all their (proper) subgroups cyclic. $\endgroup$ – Todd Leason Nov 2 '16 at 22:49

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)


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