# Classification of finite minimal non-cyclic group

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

• For those voting to close, can I ask them whether there are bounded non-zero derivations from a commutative semisimple Banach algebra to itself?
– Yemon Choi
Sep 11, 2016 at 12:19
• This question is definitely not research level in my opinion. Sep 12, 2016 at 0:48
• 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. Sep 12, 2016 at 1:27
• 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 $n > 1.$ Sep 12, 2016 at 1:38
• 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.
– Dave Witte Morris
Sep 13, 2016 at 8:10

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)

• Here is the paper: doi.org/10.1090/S0002-9947-1903-1500650-9
– lhf
Apr 15, 2019 at 17:37
• @Yanior Weg. Thank you so much for your complete answer Apr 16, 2019 at 18:27