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Let $G$ be a finite group having the property that for any prime $p$ dividing $|G|$, it has a subgroup H with $[G:H]=p$. What can be said about these groups? I believe I can prove that they must be solvable. But are they supersolvable?

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Surely, if $G$ is any finite group, then you can get a group in this class by taking the direct product of $G$ with a cyclic group of order divisible by all primes dividing $G$. –  Derek Holt Oct 4 '11 at 8:01
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$S_4$ is in this class of groups, but is not supersolvable. –  Kevin Oct 4 '11 at 8:32
    
Derek, thanks, good point! So they aren't even necessarily solvable. Class is hence very large. –  Nicky Hekster Oct 4 '11 at 8:46
    
duplicate question: math.stackexchange.com/questions/66451/… –  Julian Kuelshammer Oct 4 '11 at 11:12
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@Derek: I don't see any reason why you couldn't expand your comment into an answer. –  mixedmath Aug 21 '12 at 22:31

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