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My problem is slightly convoluted: The resource from which I am mainly following, has not introduced Orbit Stabilizer Theorem and have gone for proving it(Sylow) in a highly complex way. In short: I have lost patience with it.

However I have gone through the results and would like to apply it, to solve a few simple problems on my own.

  1. Is it advisable?
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    It seems you are aware of other proofs. Why not just read up on orbit-stabilizer and then grab a different text for the proof of Sylow? Basically any algebra text (apparently aside from yours) should do the trick.2012-09-27

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This is of course a matter of opinion, but I see no reason at all you shouldn't learn to apply Sylow's theorem before you learn to prove it. In more elementary areas of math we always do this, and the proving is almost an independent skill from the computing of Sylow $p$-subgroups.

I seem to remember a quote from the author of an analysis textbook to the effect that the reader should not tax him- or herself too much with the most technical proofs on a first pass through the material, until the layout of the whole field is understood. Dieudonne is the name that comes to mind, but I can't find a reference now.

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    I very much hope you get enough time to really absorb Sylow after your exams!2012-09-29