By "minimal nonsolvable" I mean a nonsolvable group whose all proper subgroups are solvable.
Let $G$ be a finite minimal nonsolvable group, with the following properties:
1- $G$ has only one proper normal subgroup $N$,
2- $N$ is an elementary abelian 2-group;
3- $C_{G}(N)=N$;
4- $\dfrac{G}{N}\cong A_{5}$.
5- $\forall x\in G$, $o(x)\in\lbrace 1,2,3,4,5\rbrace$.
Does there exist any group $G$ with the above conditions?
Best Answer
No such group exists.
Then assumed conditions imply that $N$ is a minimal normal subgroup of $G$ or, equivalently, the conjugation action of $G$ on $N$ gives $N$ the structure of an irreducible ${\mathbb F}_2G$-module, where ${\mathbb F}_2$ is the field of order 2.
It is well-known that there are just three such modules up to isomorphisms, $N_1,N_2,N_3$, where $N_1$ has dimension 1 and $N_2$ and $N_3$ have dimension $4$.
By Condition 3, we cannot have $N \cong N_1$. But $H^2(A_5,N_2) = H^2(A_5,N_3) = 0$, so an extension of $N_2$ or $N_3$ by $A_5$ is split, and hence it is not a minimal nonsolvable group, because it has a subgroup isomorphic to $A_5$.