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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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    @Derek: I don't see any reason why you couldn't expand your comment into an answer.2012-08-21

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