# On the finite minimal non-solvable groups

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?

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.

• Thank you so much for your answer. Would you please explain how this group is constructed in GAP? Feb 17 at 7:57
• 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 $2^2.S$ Feb 17 at 8:42
• It might be easier to use G:=AtlasGroup("2^2.Sz(8)"); from the atlasrep package. Feb 17 at 17:38
• I added a general argument, but it seems ${}^2B_2(8)$ is the only minimal simple group with suitable Schur multiplier for this question. Feb 18 at 8:31
• @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 $S$" ? Feb 24 at 9:43