Variety of limited simple groups of offered order goes to the majority of $2$ - is a classification-free evidence feasible?

This Wikipedia article states that the isomorphism sort of a limited straightforward team is established by its order, other than that:

  • $L_4(2)$ and also $L_3(4)$ both have order $20160$
  • $O_{2n+1}(q)$ and also $S_{2n}(q)$ have the very same order for $q$ weird, $n > 2$

I assume this suggests that for each and every integer $g$, there are $0$, $1$ or $2$ simple groups of order $g$.

Do we require the complete toughness of the Classification of Finite Simple Groups to confirm this, or exists a less complex means of confirming it?

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2019-05-05 11:16:38
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Answers: 1

Although I am not (by any kind of stretch) a specialist on limited simple groups, allow me expand my above comment.

Take into consideration the adhering to QCFSG (i.e., "qualitative" CFSG) : with just finitely several exemptions, every limited straightforward team has prime order, is rotating, or is just one of the finitely several well-known boundless family members of Lie type. QCFSG has to have been judged instead beforehand, whereas the specific declaration of CFSG was a lot tougher ahead by, as much of the very early work with the category trouble caused exploration of new occasional teams.

I presume that beforehand a person has to have considered the nonsporadic limited simple groups and also saw that, with the exception of both exemptions detailed above, they have distinctive orders. [Thinking this is in fact real, that is. I have no factor to question it, yet I have not examined it myself. ] As soon as you see that, if you think QCFSG, after that you absolutely assume that the order of a straightforward team establishes the team approximately finitely several exemptions . It is really tough for me to visualize just how you can confirm that the variety of exemptions is specifically 2 without recognizing the complete CFSG.

I can not stand up to sharing a tale of Jim Milne, whose ethical is that you should not really feel regrettable when you claim something definitely foolish in public : far better mathematicians than you or I have actually claimed stupider points.

Ultimately, a tale to remember the next time you ask an entirely foolish inquiry at a significant lecture. Throughout a Bourbaki workshop on the standing of the category trouble for straightforward limited teams, the audio speaker stated that it was not recognized whether a straightforward team (the beast) existed of a particular order. "Could there be greater than one straightforward team of that order?" asked Weil from the target market. "Yes, there can" responded the audio speaker. "Well, could there be definitely several?" asked Weil.

For the resource, and also for some more enjoyable tales, see

http://www.jmilne.org/math/apocrypha.html

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2019-05-08 06:46:25
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