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How do you prove that for any $n$ there are at most two simple groups of order $n$?

Julien
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sdfsadf
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    Homework? Exam? What have you tried? – John Engbers Dec 21 '13 at 02:11
  • very interesting question... It would have been better if you could have specified what you have done.... –  Dec 21 '13 at 02:19
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    http://math.stackexchange.com/questions/1423/number-of-finite-simple-groups-of-given-order-is-at-most-2-is-a-classification asks for a proof of this fact that doesn't use the classification of finite simple groups. – Julian Rosen Dec 21 '13 at 02:25
  • @JohnEngbers Your question implicitly says this is easy or routine. I'd be interested in seeing a homework-level easy/routine argument. Thanks in advance. – Julien Dec 21 '13 at 02:34
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    In this very particular case, I see no advantage in asking "what have you done". – DonAntonio Dec 21 '13 at 03:29
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    I agree that this (seems to be) a difficult question; on a question such as this I find it useful to know some context behind the posing of the question. For example, is this useful in a sdfsadf's research? Is it for a lecture in a group theory course? Eliciting further information related to the question was the reason I commented above. – John Engbers Dec 24 '13 at 19:37

1 Answers1

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You use the classification of finite simple groups. There is no a priori reason.

Igor Rivin
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    ... that we (or you?) know of... – Thomas Andrews Dec 21 '13 at 02:45
  • @ThomasAndrews What else could I mean? (to help the slow children among us: "the only proof of that fact I am aware of uses classification"). And thanks for the downvote. – Igor Rivin Dec 21 '13 at 02:54
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    "The only proof of that fact I am aware of uses classification" has a wide variety of meanings, since readers don't know the depth of your knowledge. Are you a person with deep knowledge about group theory? Etc. That's why the answer is less than useful - there is no way to tell the reliability of the claim. I suspect, from reading the answer to the related question, that you are correct, but there is not any context in your answer. – Thomas Andrews Dec 21 '13 at 02:59
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    I agree with Igor's answer: the OP is not asking for an elementary proof (alas, the OP is just spitting the question). – Matemáticos Chibchas Dec 21 '13 at 03:00
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    If I were to write "The only proof of that fact I am aware of uses classification..." that would have a different meaning than if Walter Feit wrote it. – Thomas Andrews Dec 21 '13 at 03:01
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    @ThomasAndrews You are free to Google my name and determine whether you find the information supplied trustworthy. Of course, you are also free to be a jerk, and I am glad that you are taking advantage of your freedoms. As for Walter Feit, had he posted an answer we would have deeper problems than the OPs. – Igor Rivin Dec 21 '13 at 03:04
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    But if one has to google the answerer to gauge the value of the answerer, that makes the answer not useful in the long run. I was not questioning your credentials, I was saying your answer is only useful if we know your credentials, and that makes the answer incomplete. – Thomas Andrews Dec 21 '13 at 03:05
  • As a wiser man than I had noted, in the long run we are all dead (some of us, like Professor Feit, are, unfortunately, dead in the short run, too). – Igor Rivin Dec 21 '13 at 03:07
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    So why bother answering questions at all? That's just nihilistic nonsense. If an answer's meaning depends on the level of the answerer's expertise (such as practically any "not that I know of" answer) the answer is inherently incomplete without some details of who you are. – Thomas Andrews Dec 21 '13 at 03:17
  • "You use the CFSG. There is no a priori reason"....this doesn't answer at all, or perhaps it does, the OP's question: does this mean "yes, it's true and for that use the CFSG, though there is no a priori reason"( and one could even argue here: there's no a priori reason a differentiable function has to be continuous, either: one proves that), or perhaps it means "use the CSFG, hopefully the answer is there. I've no the faintest idea, but there's no a priori reason" . About the downvoting: I don't like that under almost any circumstances. – DonAntonio Dec 21 '13 at 03:24
  • @ThomasAndrews What exactly is your point? I am not user123445, I publish my name, so anybody is free to assess my level of expertise as much as they like. – Igor Rivin Dec 21 '13 at 03:25
  • And thanks to @julien for the edit which puts an end to this ridiculous discussion. – Igor Rivin Dec 21 '13 at 03:27
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    That means that the answer is not self-contained. That's all I'm saying. The point is to give self-contained answers. See the answer to the duplicate question above. The goal of this site is to give self-contained questions and answers. Even if you included a link to your CV, I'd have complained, because the answer itself would not be complete. – Thomas Andrews Dec 21 '13 at 03:33
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    Come on guys, please stop this polemic discussion here, or do it outside this medium. – Nicky Hekster Dec 21 '13 at 09:12
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    @NickyHekster I think it was over by the time you commented. – Igor Rivin Dec 21 '13 at 15:21