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So while I've known about the classification theorem of finite simple groups for a while, I saw the orders of the sporadic groups factored and noticed that all of them have very many small prime factors. Given the importance of (and rareness of large) "smooth numbers" in various aspects of number theory, this seems to be fairly significant!

After some searches on Stack Exchange, Overflow, and Google (not digging too deep), I couldn't find anything specifically addressing this phenomenon, and having seen the Sylow Theorems and things like Burnside's Theorem, it is clear that the prime factorization is closely related to a group having/not having normal subgroups in some way.

So I wanted to ask, "Is there a (big-picture) group-theoretic explanation for why the orders of sporadic groups have no large prime factors, and how (if at all) is this at all useful in the study of these groups or other places where these groups are found?"

But, of course, the classification of finite simple groups is a several-thousand page endeavor and this question is rather broad, so I might expect a satisfactory answer to either require hundreds of pages of mathematics, or be so soft that it may be frowned upon.

So I wanted to make this post on meta to see if this question would be received well if I asked it on main.

Thoughts?

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  • $\begingroup$ Hmmm. What's tough is that I appreciate your question here, for feedback as to whether your question would "fit well" on main. For that, I'd like to upvote the question. On the other-hand, I don't think there is any way of getting around that such a question would be far too broad to ask on main (as it does not lend itself well to being answerable in the limited field provided on main, for answers. And for that reason, I'm tempted to downvote your question here on meta (which unlike on main, downvotes on meta are taken to be taken as disagreement with the proposal, request, etc. being made. $\endgroup$ – Namaste Sep 20 '17 at 22:49
  • $\begingroup$ @amWhy I would expect that there is no nice, short answer since decades (centuries?) of mathematics could likely not be summarized into a Stack-Exchange answer. Like so many of us do, I was interested and curious about a pattern I observed and was wondering if someone more well-versed than myself had a page-or-two overview of the general story/significance. I expect the truth is that such an overview is impossible, so I understand the appropriateness of a downvote. :) If you think there is an improved version of the question that could elicit a reasonable answer, I would be happy to amend it! $\endgroup$ – Christian Sep 20 '17 at 23:01
  • $\begingroup$ Christian, let me be clear, I haven't upvoted, nor downvoted. I was merely addressing a question very common to many, if not most, meta post: what any one user's downvote on a meta post might mean is anyone's guess, for example, what I describe. I said that mainly to ensure that you understand that downvotes are very unlikely to reflect that you shouldn't have asked the question on meta, and that any downvotes would likely only convey something to the effect: "I don't think this is a good idea for posting such a question on main." It was as much a comment to other users as it was to you. $\endgroup$ – Namaste Sep 20 '17 at 23:08
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    $\begingroup$ But given your comment, Christian. Such a question on main might be successful if you phrased it (and tagged it) as a reference request, with the description of what interests you, as you describe above, and a brief summary of where you are in your study of groups (so as to help potential answerers better recommend references that will best meet you where you're at, and help guide you forward. From your post, it is clear that you've thought a lot about this question, so I think it could be a successful, well-received question on main. $\endgroup$ – Namaste Sep 20 '17 at 23:12
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With a bit of additional work I think this would be suitable to post on the main site.

The additional work I have in mind is to list the sporadic simple groups and their largest prime divisor of their order, possibly along with another measure of the smoothness of these orders.

By the standard of "shows research" this should be adequate (even interesting to some). It is no guarantee of an answer, of course.

I asked a Question once in a similar style, providing some numerical counts to support it. It earned me a Tumbleweed Badge.

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