## Groups that do not exist

In the long process that resulted in the classification of finite simple groups, some of the exceptional groups were only shown to exist after people had computed (most of) their character tables and other such precise information which usually can only be attached to things that exist. Maybe someone familiar with the details can tell me/us:

Was there at some point a finite simple group conjectured to exist that turned out not to exist in the end?

If so, this part of the story is much less told than the successful part! It would be interesting to know if for such a non-existent group, say, the character table was computed, and so on...

I am asking because I just read this question on Math.SE and it reminded me I have always wanted to know this.

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A natural follow-up question, of course, is if, say, the character table of that group which did not exist is the character group of something else. – Mariano Suárez-Alvarez Dec 7 at 19:11
I really prefer for answerers to be rewarded for their effort :-) – Mariano Suárez-Alvarez Dec 7 at 19:34
@36min: The $p^aq^b$ theorem is due to Burnside, not Frobenius... – Arturo Magidin Dec 7 at 20:45
The Illuminati. – Will Jagy Dec 7 at 20:48
We have managed to convince you that we do not exist Will, good. ... Hmm. Oops. – Mariano Suárez-Alvarez Dec 7 at 21:01

There was a point during the history of the Classification when pursuers of sporadic groups distinguished the Baby Monster, the Middle Monster and the Super Monster. The first two actually turned out to exist (though the word "Middle" was dropped), but the third turned out to be a dud.

http://www.neverendingbooks.org/index.php/tag/simples/page/2 has an account of this.

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Good example. I believe that it's the Middle Monster that failed to exist. According to Wikipedia, the automorphism group of Fi$_{22}$ centralizes an element of order 3 in the baby monster, and there's a triple cover of Fi$_{24}$ that centralizes an element of order 3 in (what is now called) the monster. There is no analogous fact for Fi$_{23}$. – Timothy Chow Dec 8 at 21:44
The permalink would be neverendingbooks.org/index.php/… – Harry Altman Dec 8 at 22:23
Ah! This is exactly what I wanted. Does the non-existent one exist in a non-groupy way, perchance? – Mariano Suárez-Alvarez Dec 9 at 2:06

I tried to write a longer answer which froze, so I'll write a shorter version. You might look at the history of the "Solomon fusion system" which arose in a characterization problem undertaken by Ron Solomon in his work on the classification of finite simple groups. This does not occur in a finite group, but was shown by Dave Benson to occur in a group like topological object ( "2-adic loop space") called BDI(4).

In some sense this led to work by topologists (especially Broto, Levi and Oliver) on "$p$-local finite groups" (actually topological spaces, not groups) which need to associate a linking system to a fusion system of a finite $p$-group. Aschbacher and Chermak showed in an Annals paper a few years ago that the Solomon fusion system does have an associated linking system, an therefore there is a $2$-local finite group associated to that fusion system. More recently, Chermak has shown that there is a $p$-local finite group associated to every saturated fusion system on a finite $p$-group.

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 I'll look this up. Thanks! This sounds like right what had in mind :-) – Mariano Suárez-Alvarez Dec 8 at 14:28

I'm not sure if this is quite what you're looking for but....

In this book "Finite simple groups", Gorenstein tells the story of Feit & Thompson's proof of the odd order theorem. Very roughly, it goes as follows:

Suppose $G$ is a simple group of odd order. Thompson studied the local structure of the group $G$ to obtain information about the structure of the maximal subgroups of $G$. Feit then applied the Brauer-Suzuki theory of exceptional characters to derive a great deal of character-theoretic information about the group $G$. So far so good.

But now they hit a problem. They were seeking, of course, to demonstrate a contradiction. But, as Gorenstein tells it, one of the possible configurations of maximal subgroups & character information proved extremely difficult to disprove. In the spirit of this question, one might say they found an example of a "group that does not exist". In the end, after spending a year being stuck, Thompson managed to demonstrate the required contradiction by a very delicate analysis of the generators and relations of the putative group $G$.

(I don't have a copy of Gorenstein's book with me. If I get chance I might return to this answer so I can provide some quotes. Gorenstein's account of the whole enterprise is really terrific.)

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Yes `the conjectured to exist' part of your question made me wonder if the FT-theorem was relevant. Still, as Gorenstein tells it, there is a real sense that something awfully like a group was lurking in that final configuration... Maybe a group that wanted to exist but didn't know how :-) – Nick Gill Dec 7 at 21:36
Is there a formal sense in which "there is almost such a group"? $\:$ (e.g., a binary operation on a set such that associativity fails for exactly one ordered triple of elements, which is otherwise a group) – Ricky Demer Dec 7 at 21:45
On p. 212 of Dummit & Foote (3rd ed.), the possibility of a finite simple group of order $3^3 \cdot 7 \cdot 13 \cdot 409$ is discussed, with a certain amount of consistent data for such a group being obtained, but ultimately there can't be such a group since there are no simple groups of odd order. – KConrad Dec 7 at 23:20
I'm quite sure I can construct an infinite sequence of simple groups of odd order... – Mark Meckes Dec 8 at 13:08
It's common practice among group theorists to omit the locution "except for cyclic groups of prime order" for brevity, understanding that the listener will insert it wherever it is necessary. – Timothy Chow Dec 9 at 0:35