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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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The chapters/topics before the topic covering Sylow THeorem doesnt cover Orbit Stabilizer Theorem. Without this tool, they have gone for proving Sylow theorem in a highly convoluted way –  Soham Sep 27 '12 at 0:46
    
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. –  Mike B Sep 27 '12 at 0:54
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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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While a proof isn't necessary to apply the theorem, understanding a proof can be very helpful in applying the theorem appropriately. I recall several instances of that when I was doing topology, I'll see if I can dig up a few examples. –  Michael Dyrud Sep 27 '12 at 1:59
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I absolutely expect there will be good cases when one actually uses the technique of the proof in making applications. I think Sylow's theorems have a particularly wides gap between proof and practice, though-they're more like the equivalence of singular and simplicial homology in topology than like the standard techniques of proving property X is preserved by products, which are important to know for applications as well. –  Kevin Carlson Sep 27 '12 at 2:03
    
@KevinCarlson Thanks though this is a music to my ears, but I feel guilty for this. A deadline (euphemism for exams) is looming over my head else, I can really appreciate and understand the beauty of group theory. –  Soham Sep 29 '12 at 18:29
    
I very much hope you get enough time to really absorb Sylow after your exams! –  Kevin Carlson Sep 29 '12 at 23:01
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