Let $ G $ be a minimal non-solvable group. Why $ G/\Phi(G) $ is a minimal simple group, where $ \Phi(G) $ is the Frattini subgroup of $ G $ ?

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    $\begingroup$ This answer says that a minimal non-solvable group is simple (actually it depends on the definition). What is your definition of minimal non-solvable group and minimal simple group? $\endgroup$ – Crostul Oct 12 '15 at 20:40
  • $\begingroup$ @Crostul But that question was wrong! As is proved in one of the answers, ${\rm SL}(2,5)$ is minimal non-solvable (i.e. all of its proper subgroups are solvable), but it is not simple. $\endgroup$ – Derek Holt Oct 12 '15 at 22:00

If $G/\Phi(G)$ is not simple, then let $N/\Phi(G)$ be a minimal normal subgroup. Then $N < G$, so $N$ is solvable. Let $M$ be a maximal subgroup of $G$ not containing $N$ (such an $M$ must exist since $N$ is not contained in $\Phi(G)$). Then $MN=G$, and since $N$ is solvable but $G$ is not, $M$ cannot be solvable, contradicting $G$ being minimal non-solvable.

So $G/\Phi(G)$ is simple and is clearly minimal simple.


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