Is it possible for a group (non-simple and non-abelian) that solvability of all of its proper subgroups leads the whole group to be solvable?
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1$\begingroup$ Minor edits. In any case, the question may be too elementary for MO. $\endgroup$– Jim HumphreysCommented Mar 25, 2011 at 12:27
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1$\begingroup$ Sorry Jim for my elementary questions. $\endgroup$– MikasaCommented Mar 25, 2011 at 12:32
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3 Answers
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No. $SL(2,5)$ is a non-simple non-solvable group with the property that all its proper subgroups are solvable.
answered Mar 25, 2011 at 8:54
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3$\begingroup$ And that's the smallest example. Also called the binary icosahedral group. Maps onto $A_5$ with kernel of order $2$. $\endgroup$ Commented Mar 25, 2011 at 11:06
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An even simpler counter example is $A_5$.
I believe that finite simple groups in which every proper subgroup is solvable are called minimal finite simple groups and as I recall they were classified by J. Thompson before the calssofication of all finite simple groups. This classification is useful, I think J. Wilson used them to study identities of solvable groups.
answered Mar 25, 2011 at 9:09
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3$\begingroup$ The OP asked for a non-simple group. $\endgroup$ Commented Mar 25, 2011 at 10:14
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$\begingroup$ J. Thompson, in his (famous) series of papers, dealt also with not necessarily simple groups, as far as I remember. $\endgroup$ Commented May 8, 2011 at 10:38
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The minimal non-solvable group surely has the property that all proper subgroups are solvable.
answered Mar 26, 2011 at 18:19
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2$\begingroup$ The only (very slight) subtlety is that the OP asked for a non-simple example. $\endgroup$– HJRWCommented Mar 26, 2011 at 22:16