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I need some help proving this proposition.

"$G$ is a solvable group if and only if $G$ has a Sylow system"

(Sylow system: a set $S$ of Sylow subgroups of $G$, one for each prime dividing $|G|$, so that if $P$, $Q$ $\in{S}$, then $PQ=QP$).

Thanks a lot!

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The one direction of this, if I am not mistaken, is just Hall's theorem. Perhaps you should look at a proof of that. – Alex Youcis Dec 10 '12 at 13:20
Well, this is a rather serious and pretty fat theorem, sometimes known as Hall's Theorem (there are several results that go by this name). You better grab some good group theory book in your university's library, or try to google. You can also check from page 81 (this like page 16 in the document's numbering) in – DonAntonio Dec 10 '12 at 13:21
Read 9.1.7 , 9.1.8 in Robinson's "A Course in the Theory of Groups" (page 257, 2nd Edition 1996) – DonAntonio Dec 14 '12 at 15:59

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