# 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 '16 at 12:19
• This question is definitely not research level in my opinion. – Geoff Robinson Sep 12 '16 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. – Geoff Robinson Sep 12 '16 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 $n > 1.$ – Geoff Robinson Sep 12 '16 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 '16 at 8:10

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 $$r ≡ 1 \pmod q$$, $$r^q ≡1 \pmod p$$.