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I attended a talk by John Conway recently where he explained to us that the usual number, 26, was wrong, that there are in fact 27 sporadic simple groups. His reason was that the Tits group, which is often claimed to be a group of Lie type is not.

The reasoning seems pretty sound; it is usually lumped in with the groups ${}^2F_4(2^{2n+1})$, but these are only simple for $n \geq 1$: ${}^2F_4(2)$ is not simple.

The Tits group is an index 2 subgroup of this group, and it is simple. So it is not of this form, no matter how "almost" it is. Moreover, these groups all have a BN-pair, while the Tits group does not.

So to me it seems clear that there are 27 sporadic groups, not 26.

What I'd like to understand is this: What is the reasoning to include it in the infinite family ${}^2F_4(2^{2n+1})$? There must be some reasons that people do list it as a member of that family other than surface similarity.

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    $\begingroup$ It's included in that family because it's equal to $^2F_4(2)'$. So it's not surprising that people don't agree about whether it's sporadic or not. $\endgroup$
    – Derek Holt
    Dec 10, 2013 at 18:20
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    $\begingroup$ I remember noticing at some point in the 1970s that the number of known sporadic simple groups was equal to the number of known Mersenne primes. Since then, the group theorists have fallen way behind the number theorists. $\endgroup$
    – Gerry Myerson
    Dec 10, 2013 at 22:23
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    $\begingroup$ I don't mean this in a confrontational way at all, but: does it really matter whether we view this group as sporadic or not? If so, why? (By way of comparison, Conway likes to talk about "the prime $-1$" rather than the infinite prime. I am confident that it doesn't matter whether we say it that way or not.) $\endgroup$
    – Pete L. Clark
    Dec 11, 2013 at 0:51
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    $\begingroup$ Anyway, the point of my remark is that (so far as I see it, obviously) the classification theorem for finite simple groups is not really "There are precisely $18$ infinite families and $26$ sporadic simple groups." It is "Although there are infinitely many finite simple groups, we can describe them in a very explicit, finite way: namely..." Even the five families have some overlap...but that's just a foible of our description of the classification, it seems to me. $\endgroup$
    – Pete L. Clark
    Dec 11, 2013 at 5:17
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    $\begingroup$ I can't see how you're going to get a definitive answer to this question - it all depends on what you mean by 'sporadic'. Perhaps it would be as well to say that certain groups 'exhibit sporadic behaviour', rather than 'are sporadic'. For example, as Noam says, $A_5$ has several other incarnations both of which are associated with a BN-pair. So $A_5$ exhibits sporadic behaviour because it is alternating AND has two different BN-pair structures! Is that more or less sporadic than ${^2F_4}(2)'$ which is of Lie type but has no BN-pair? Depends on who's asking I guess... $\endgroup$
    – Nick Gill
    Dec 11, 2013 at 14:12

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