How surprised would you be if mathematicians discovered a 27th sporadic finite simple group?


My answer to question on Quora: How surprised would you be if mathematicians discovered a 27th sporadic finite simple group?

I would be really surprised.

I am one of the few people in the world who had reason to read, and spent some time reading, certain parts of the proof of the Classification of Finite Simple Groups (CFSG), and I have some basic understanding of what is going on. A few points:

  1. Many parts of the proof of the CFSG (in its various versions) are done by induction, by considering a minimal counterexample: a smallest, by order, finite simple group which is not on the list. The arguments involved are fine tuned at identification of a new finite simple group, if one exists. The fact that this has not happened in the last 30 year is quite reassuring.
  2. Some “classical” groups are more sporadic (in the sense that their properties are quite abnormal) than most sporadic group (a good example is the projective special linear group \(PSL_3(4)\) which should be seen as Mathieu group   \(M_{21}\). In that sense there are already more than 26 sporadic simple groups.
  3. I heard some good mathematicians suggesting that at least some of the sporadic groups are likely to belong to infinite series of a new kind of algebraic structures still unknown to us, but some of which have happened, by chance, be groups — the same way as the alternating group \(Alt_6\) has happened to be the linear group \(PSL_2(9)\), living simultaneously in two different universes.

The third point, if confirmed, would be really exciting and likely to have long lasting impact on mathematics.

I heard the conjecture the “sporadic groups are not groups”, from Israel Gelfand, among other people. Gelfand was quite keen to develop it, and returned to it again and again in conversations with me and with others (this was in 1990s – early 2000s).

There is a tantalising example of the Mathieu pseudogroup \(M_{13}\) proposed by John Conway, and a few authors were trying to expand this line of inquiry. A good discussion of Conway’s \(M_{13}\) and bibliography can be found in Ben Fairbairn’s paper Some Examples Related to Conway Groupoids and their Generalisations.f

Finally, there is a general classification scheme for finite structures, the Cherlin-Lachlan theory — I’ll try to write a few words on it later.