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By minimal non-solvable group, I mean a non-solvable group whose all proper subgroups are solvable. Let $G$ be a minimal non-solvable group. If $G$ is not a simple group, then it can be easily verified that $G$ contains only one maximal normal subgroup say $N$. Moreover, in this case we can prove that for each nontrivial normal subgroup $H$ of $G$, we have $N\cap H=N$ or $H$, since otherwise $NH=G$ which contradicts nonsolvablity of $G$. More precisely, any other possible proper normal subgroup of $G$ is incuded in $N$. Now my question:

Does there exist any example of a minimal nonsolvable group $G$ such that $G$ has two nontrivial proper normal subgroups?

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A natural place to look for examples would be to look for a minimal simple group with non-cyclic Schur multiplier.

The Schur multiplier of the Suzuki group $S = {}^2B_2(8)$ is elementary abelian of order $4$.

It is possible to verify (for example by constructing this group in GAP or Magma) that the covering group $G = 2^2.S$ is minimal non-solvable. (Probably there are non-computational ways to do this which are not too difficult)

In any case here the center is elementary abelian of order $4$, which gives you four nontrivial proper normal subgroups.

EDIT: Here is a general argument. Let $S$ be a finite simple group, and let $G$ be a perfect central extension $$1 \rightarrow Z \rightarrow G \rightarrow S \rightarrow 1.$$ Consider a maximal subgroup $M < G$. If $Z \not\leq M$, then $G = MZ$. Since $Z$ is central, then $M$ is normal. But then $M$ being a maximal subgroup implies $G/M$ is cyclic of prime order, contradicting the assumption that $G$ is perfect.

So every maximal subgroup $M < G$ satisfies $Z < M < G$ and $M/Z$ is maximal in $G/Z \cong S$.

Therefore if $G$ is a perfect central extension of a minimal simple group $S$, then $G$ is a minimal non-solvable group.

Then for the examples that the question asks for, just find a minimal simple group $S$ such that the Schur multiplier is not cyclic of prime order, and not trivial.

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  • $\begingroup$ Thank you so much for your answer. Would you please explain how this group is constructed in GAP? $\endgroup$ Feb 17 at 7:57
  • $\begingroup$ You can find generators for $2^2.S:3$ here: brauer.maths.qmul.ac.uk/Atlas/v3/permrep/4Sz8d3G1-p2080B0 Take for example the derived subgroup of this to get $2^2.S$ $\endgroup$ Feb 17 at 8:42
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    $\begingroup$ It might be easier to use G:=AtlasGroup("2^2.Sz(8)"); from the atlasrep package. $\endgroup$ Feb 17 at 17:38
  • $\begingroup$ I added a general argument, but it seems ${}^2B_2(8)$ is the only minimal simple group with suitable Schur multiplier for this question. $\endgroup$ Feb 18 at 8:31
  • $\begingroup$ @testaccount. By what you explained in the above, is it true to say that "If $G$ is a nonsimple group and minimal nonsolvable then $G$ is a central extension of a finite nonabelian simple group $S$ by an elementary abelin group say $E$ where the order of $E$ divides Schur multiplier of $S$" ? $\endgroup$ Feb 24 at 9:43

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