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By "minimal non-solvable group" I mean a non-solvable group whose all proper subgroups are solvable.

I need a proof or a counterexample for the following proposition:

Let $G$ be a finite minimal non-solvable group. Then one of the following occurs:

(a) $G$ is a finite non-abelian simple group;

(b) $G$ has a prime order normal subgroup $N$, such that the quotient group $\dfrac{G}{N}$ is a finite non-abelian simple group.

Thank you in advance

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  • $\begingroup$ @ dodd, Could you please give a reference,(book or paper). $\endgroup$ – H.Shahsavari Oct 14 '20 at 20:40
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It's not true. A counterexample is a nonsplit extension $2^3.L_3(2)$ of an elementary abelian group of order $8$ by the simple group ${\rm SL}(3,2)$.

Then $G$ has a normal subgroup $N$ with $G/N$ simple, but $N$ does not have prime order.

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  • $\begingroup$ @ Derek Holt. Thank you so much for the answer. Do you have any suggestion to correct part (b) in the proposition? $\endgroup$ – H.Shahsavari Oct 14 '20 at 21:42

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