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I have seen two proofs of the simplicity of $A_n,~ n \geq 5$ (Dummit & Foote, Hungerford). But, neither of them are such that they 'stick' to the head (at least my head). In a sense, I still do not have the feeling that I know why they are simple and why should it be 5 and not any other number (perhaps this is only because 3-cycles become conjugate in $A_n$ after $n$ becomes greater than 4).

What is the most illuminating proof of the simplicity of $A_n,~ n \geq 5$ that you know?

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    The appear in the list of the classification theorem! :)2010-12-28
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    "Most illuminating" lies in the eyes of the beholder. This is a very subjective question, to which there is no correct answer.2010-12-29
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    @Alex: I agree it's subjective, but not in the specific pejorative sense the word is used on this site. I think it's certainly worthwhile to see a collection of summaries of proofs of this result and have the OP select one and explain why it fits his/her needs best. (Full disclosure: I never remember how to prove the simplicity of $A_n$ either, in part for lack of trying. So I see the possibility of personal edification here.)2010-12-29
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    At http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/Ansimple.pdf there are five proofs given for the simplicity of A_n when n is at least 5 (the proofs in Dummit & Foote and in Hungerford are proofs 1 and 4). Maybe you'll like one of the arguments given there more than the ones you have seen already.2010-12-30

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