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I am looking for a complete classification of minimal finite non-cyclic groups. Is there any paper or book?

Chain Markov
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H.Shahsavari
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    For those voting to close, can I ask them whether there are bounded non-zero derivations from a commutative semisimple Banach algebra to itself? –  Sep 11 '16 at 12:19
  • @YemonChoi I guess what you are saying is that you disagree with the votes to close! I think the votes are to transfer the question to MSE on the grounds that it has already been answered there. But I will withdraw my vote to close anyway. (And I am afraid that I don't know the answer to your question) – Derek Holt Sep 11 '16 at 15:29
  • @DerekHolt It was, IIRC, a not unreasonable Part III exam question at some point in the late 1990s, ergo not research level :) To be fair, it did come with hints. –  Sep 11 '16 at 16:01
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    This question is definitely not research level in my opinion. – Geoff Robinson Sep 12 '16 at 00:48
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    The possibilities are: a quaternion group of order $8$, a direct product of two cyclic groups of order $p$ for some prime $p$, or a group of order $pq^{n}$ where $p$ and $q$ are primes with $p$ congruent to $1$ (mod $q$), there being one possibility up to isomorphism for every positive integer $n$, which has cyclic subgroups of order $pq^{n-1}$ and $q^{n}$ and has only one Sylow $p$-subgroup. – Geoff Robinson Sep 12 '16 at 01:27
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    Here I mean noncyclic groups with all proper subgroups cyclic. If you want groups which also have every proper homomorphic image cyclic, then the quaternion group of order $8$ should be omitted, as should the last type in case $n > 1.$ – Geoff Robinson Sep 12 '16 at 01:38
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    This was posed purely as a question in group theory. In that setting, it is clearly not research level. It could be evaluated differently if, say, the OP explained that it arose in the study of commutative semisimple Banach algebras. –  Sep 13 '16 at 08:10
  • A related MO post: Group in which all subgroups are cyclic. – Martin Sleziak Apr 16 '19 at 13:52

1 Answers1

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A theorem by G.A.Miller and H.C.Moreno states:

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 $r ≡ 1 \pmod q$, $r^q ≡1 \pmod p$.

This theorem first appeared in "Non-abelian groups in which every subgroup is abelian" by G.A.Miller and H.G.Moreno (1903)

Chain Markov
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