If $H$ and $K$ are solvable subgroups of G with $H$ normal in $G$, then $HK$ is a solvable subgroup of $G$.
I know a group is solvable if and only if it has a solvable series. But I don't know how to construct the solvable series of $HK$.
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haigang hu
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@rschwieb ,thanks for your link, and I find my question is no false, the link you give to me have already solved my question. My question is found in book "Algebra" by Tomas W. Hungerford. – haigang hu Jun 18 '15 at 17:17
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@rschwieb If you look that link you will find that answer there is a counterexample to the question posed rather than a proof! – Derek Holt Jun 18 '15 at 17:52