Groups that do not exist - MathOverflow

Groups that do not exist

2012-12-07T19:08:58Z

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.

Answer by Nick Gill for Groups that do not exist

2012-12-07T21:18:56Z

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.)

Answer by Geoff Robinson for Groups that do not exist

2012-12-08T11:15:23Z

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.

Answer by DavidLHarden for Groups that do not exist

2012-12-08T20:49:46Z

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.