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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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Minor edits. In any case, the question may be too elementary for MO. –  Jim Humphreys Mar 25 '11 at 12:27
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Sorry Jim for my elementary questions. –  Babak S. Mar 25 '11 at 12:32

3 Answers 3

up vote 14 down vote accepted

No. $SL(2,5)$ is a non-simple non-solvable group with the property that all its proper subgroups are solvable.

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And that's the smallest example. Also called the binary icosahedral group. Maps onto $A_5$ with kernel of order $2$. –  Tom Goodwillie Mar 25 '11 at 11:06

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.

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The OP asked for a non-simple group. –  Zev Chonoles Mar 25 '11 at 10:14
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Oops you are right. –  Yiftach Barnea Mar 25 '11 at 11:03
    
J. Thompson, in his (famous) series of papers, dealt also with not necessarily simple groups, as far as I remember. –  Pasha Zusmanovich May 8 '11 at 10:38

The minimal non-solvable group surely has the property that all proper subgroups are solvable.

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The only (very slight) subtlety is that the OP asked for a non-simple example. –  HJRW Mar 26 '11 at 22:16

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