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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?

edited Mar 25 2011 at 12:26

Jim Humphreys
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asked Mar 25 2011 at 8:47

Babak Sorouh
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Minor edits. In any case, the question may be too elementary for MO. – Jim Humphreys Mar 25 2011 at 12:27

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Sorry Jim for my elementary questions. – Babak Sorouh Mar 25 2011 at 12:32

Tom De Medts · Accepted Answer · 2011-03-25 08:54:05Z UTC

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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

Tom De Medts
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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 2011 at 11:06

Yiftach Barnea · Answer 2 · 2011-03-25 09:09:20Z UTC

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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

Yiftach Barnea
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The OP asked for a non-simple group. – Zev Chonoles Mar 25 2011 at 10:14

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Oops you are right. – Yiftach Barnea Mar 25 2011 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 2011 at 10:38

Péter Komjáth · Answer 3 · 2011-03-26 18:19:05Z UTC

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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

Péter Komjáth
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The only (very slight) subtlety is that the OP asked for a non-simple example. – HW Mar 26 2011 at 22:16

About solvable groups

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