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I do know that there is a "long proof" (I have read that it is +1000pages) that the Monster group is the largest sporadic simple group. My question is:

Why can we be sure that there is no other bigger group out there?

Beyond that long proof, is there any alternative "proof" of why the Monster group is the largest simple group? After all, the full set of sporadic simple groups seems to have some unexplained and yet mysterious relationships between some of them.

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    $\begingroup$ No one to date has, as far as I know, given an a priori reason why there should only be finitely many sporadic simple groups. It seems though that there are. I suppose that if you did not believe the classification of finite simple groups, and wanted some "intuitive" explanation, it might be that to describe a new infinite collection of "sporadic" simple groups, you would have to have some parametrization or other uniform way of describing them, so you would introduce a new family of "known" groups. Almost by definition, only finitely many "sporadic" groups could be outside "known" families. $\endgroup$ – Geoff Robinson Apr 15 '13 at 15:46
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    $\begingroup$ The Monster is not the largest simple group. It is not even the largest non-abelian simple group. It is merely the largest sporadic simple group. I believe, for example, that $E_{11}$ is larger. $\endgroup$ – user1729 Apr 15 '13 at 15:55
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    $\begingroup$ @riemannium $E_{11}$ is what I came up with when I tried to think of $E_8$. $\endgroup$ – user1729 Apr 15 '13 at 16:05
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    $\begingroup$ @user1729 You can find examples of groups a billion times bigger than the monster group in any of the infinite families of finite nonabelian simple groups. I mean, hell, just take $A_{|M|}$, where $M$ is the Monster. $\endgroup$ – Alexander Gruber Apr 15 '13 at 16:17
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    $\begingroup$ It's 10000 pages, by the way, not 1000, and even that may be a conservative estimate. $\endgroup$ – Geoff Robinson Apr 15 '13 at 16:44