Memories of CFSG

Rebecca Waldecker is collecting material for a history of the Classification of Finite Simple Groups (CFSG). Prompted by this, I wrote down a few thoughts.

Although I was never involved in any project which fed directly into CFSG, I was on the edge of it for a good part of my career. What follows are some notes about this.

Student days

When I arrived in Oxford as a DPhil student in 1968, I had been assigned to Graham Higman; but since he was away on leave at the time, I started with Peter Neumann. When Higman returned, he decided fairly soon that I would be better off with Neumann. Probably, if I had continued with Higman, I would have been more involved with CFSG, since a number of Higman’s students had worked on “odd characterisations” or constructions of specific simple groups.

Peter Neumann, on the other hand, was happy for me to find my own direction (after initially telling me to read Wielandt’s Finite Permutation Groups), although he was full of good ideas whenever I had a question.

In my thesis I worked on primitive permutation groups in which the point stabiliser acts 2-transitively on some orbit. The graph-theoretic and combinatorial methods introduced by Donald Higman and Charles Sims in the 1960s proved to be excellent tools for this job. The Higman–Sims group, and the graph which we then called the Higman–Sims graph (which we found out later had been discovered by Dale Mesner, though he never investigated its automorphism group) played a starring role in my thesis.

At that time, sporadic simple groups were popping up all over the place. Inevitably I spent some of my time looking for combinatorial objects which might admit such a group, though I never came close to finding one. I do recall walking along a street in London, when I passed a van whose side proclaimed “The Cameron Group”, followed by a seven-digit number; sad to say, it was an odd number.

One person who had examined many possibilities was Arunas Rudvalis, who struck gold just once. Many years later, I met him and we talked about this time. One example that stuck in his mind was the following. The Mathieu group M₂₂ acts on the cosets of a subgroup A₇ as a rank 3 permutation group of degree 176. Various indications suggest that there might be a 2-transitive overgroup; indeed this turned out to be the case, when Graham Higman discovered the 2-transitive action of a finite simple group (which later was shown to be the Higman–Sims group). Rudvalis had observed that there is a similar rank 3 action of the Fischer group F₂₂ on the cosets of a subgroup O₇(3), and similar combinatorial indications suggest that it too might have a 2-transitive overgroup. The existence of such an overgroup was disproved using CFSG, but this seemed unsatisfactory to us. So we dug deeper into the geometry of F₂₂ and were able to give a direct proof of the nonexistence of the possible 2-transitive overgroup, in our joint paper [5].

Towards the infinite

In the late 1970s, there was a feeling that CFSG was nearing a conclusion, and people’s thoughts turned to what to do next. I was already at least as much a combinatorialist as a finite group theorist, so had no need to make detailed escape plans. But an interesting new direction presented itself when John McDermott visited Oxford and gave a seminar on infinite permutation groups. He observed that the group of order-automorphisms of the rationals (or, indeed, various other ordered sets) was k-homogeneous (or k-set transitive) for all positive k but not even 2-transitive. He gave one example of a group which was k-homogeneous for all k and 2- but not 3-transitive: the order preserving or reversing permutations of the rationals. He had constructed a very different example, using Shult’s Graph Extension Theorem, a result about finite groups which had led indirectly to the Buekenhout–Shult theorem classifying finite-rank polar spaces. Then he challenged us to find whether there were any more.

I took up the challenge, which led to my first result on infinite permutation groups [2]. I was able to re-interpret McDermott’s last example as the group of order-preserving permutations of a circular order (which could be regarded as a transitive extension of the first example, bending the line around and plugging the gap with a point to form a circle, which could be regarded as the complex roots of unity). Then it was straightforward to find an example which is k-homogeneous for all k and 3- but not 4-transitive (preserve or reverse the circular order). Then I set out to prove that there were no more. This was a struggle, using the methods I was familiar with: finite group theory and combinatorics.

After the paper was published, Graham Higman, and independently the model theorist Wilfrid Hodges, found completely different and probably better proofs, using tools from logic (the Compactness Theorem) and infinite combinatorics (Ramsey’s theorem). This sparked my interest in logic—I wanted to be able to use these tools—and subsequently I lectured on logic in many different places.

Following this, I moved on to study oligomorphic permutation groups, those with only finitely many orbits on k-sets for all k. (I was responsible for introducing the name, after consulting a neighbour who translated modern Greek broadcasts for the BBC). Though I didn’t realise it at the time, these groups of countable degree (or more precisely, their closures, in the topology of pointwise convergence) are precisely the automorphism groups of the countably categorical structures, those which are determined up to isomorphism by their first-order theory and the condition of countability.

Other finite group theorists took a different route; several of those in Oxford went into University or College administration.

O’Nan–Scott and consequences

In 1979, a big conference on finite groups in Santa Cruz was thought of by many as a chance to take stock and see whether the theorem had been proved; but it was much more than that.

I was not at the conference, but I had a sabbatical term in 1980, which I spent at the University of Sydney. Terry Gagen, a friend in the pure mathematics department, had been at Santa Cruz, and brought back a copy of the preliminary proceedings, which he was happy for me to borrow.

The proceedings contained many interesting things, but for me the highlight was a theorem stated independently by Mike O’Nan and Leonard Scott in their papers, classifying maximal subgroups of symmetric groups by their socles. In fact, the intransitive, and the transitive but imprimitive, maximals are easy to describe (as subset or partition stabilisers in the symmetric group), so the interesting new information was really about the structure of primmitive permutation groups. This subject had been intensively studied since at least the time of Jordan, and while the theorem of O’Nan and Scott appeared to be quite elementary, it appeared that it had not previously been known. The closest approach was a theorem of Burnside saying that a 2-transitive finite permutation group has a unique minimal normal subgroup which is either elementary abelian and regular, or primitive and simple.

This brings me to two historical puzzles which I have not been able to resolve. The first is: was it really new? A few years later, I was discussing this with Peter Neumann in his office. He went to his bookshelf and reached down his copy of Jordan’s Traité des Substitutions [6], and opened it to a page where there was a theorem which looked very similar to what O’Nan and Scott had done. I didn’t study it closely enough to say this with complete confidence, but I had the impression that the essential part of O’Nan and Scott was already in there.

This leads on to the second mystery: why had it lain undiscovered for so long? The only answer I can suggest is that, while the theorem is elementary, it is not very useful for permutation group theory without a lot more information about the simple groups than was then available. (The socle of a primitive group is a direct product of isomorphic simple groups, and the most intractible case is when there is only one factor. One needs detailed knowledge about the subgroups and representations of groups which are close to simple in order to apply it usefully.)

A subsidiary mystery is: why did O’Nan’s paper disappear from the published proceedings of the conference, and only Scott’s [7] remain?

Anyway, having time on my hands in Sydney, I offered a course of lectures on consequences of CFSG for finite permutation group theory. Once I started investigating this, I realised what a powerful technique had been placed in our hands. Here is just one example. The history of the orders of primitive groups has a long history, having been the 1870 Grand Prix of the Paris Academy (both Jordan and Mathieu entered submissions, but the prize was not awarded). After a number of results, Wielandt succeeded in showing that a unniprimitive (primitive but not 2-transitive) subgroup apart from S_n and A_n has order at most 24ⁿ; this was improved to 4ⁿ (proved for all primitive groups) by Praeger and Saxl. In the early 1980s came a bombshell from László Babai, who proved a bound n^4n1/2log n for uniprimitive groups, using techniques from combinatorics and probability theory.

However, what I discovered was that, assuming CFSG, and knowledge then available about simple groups, we could reduce the bound to n^c log n with “known” exceptions, and even to n^{c log log n} if you were prepared to allow more (and not so precisely defined) exceptions.

At Peter Neumann’s suggestion, I wrote this up as a survey article for the Bulletin of the London Mathematical Society, where it was published in 1981 [3]. It is now my most highly cited paper.

After this, I was invited to join many collaborations using CFSG to answer seemingly difficult questions about finite permutation groups. The most successful of these was the paper with Cheryl Praeger, Jan Saxl and Gary Seitz, proving the \emph{Sims conjecture [4]. This conjecture asserted the existence of a function f such that, if the point stabiliser G_α in a finite primitive permutation group has a non-trivial orbit of size d, then |G_α| ≤ f(d).

This activity did have one unwanted side effect. More than once, a group theorist accused me of stealing the glory which should rightfully belong to those who had proved CFSG.

Subsequently, at the invitation of Cheryl Praeger and Csaba Schneider, I returned to an aspect of O’Nan–Scott, and gave a characterisation of the geometry of diagonal groups similar to their work on product action groups (the difference being that we used partitions rather than subsets) [1].

Problems

What else would I like to see done?

Arising from the orders of primitive groups, I conjectured that any primitive group except for “known” families has a base of size at most 7 (with M₂₄ as the unique extreme case). This was proved in a sequence of papers by Tim Burness and various collaborators.

There is a trivial proof by Jordan that a transitive permutation group of degree greater than 1 contains a fixed-point-free element. In the 1980s, Fein, Kantor and Schacher proved that it is possible to choose such an element with prime power order (though not with prime order). Their proof uses CFSG in an essential way. I feel strongly that this should be unnecessary; there should be a CFSG-free proof. But there is a related conjecture which is still open and I would like to see proved: if n = p^ab where p does not divide b and a is sufficiently large compared to b, then there should be a fixed-point-free element of p-power order. Even the first case of this conjecture, for the prime 2, made by John Isbell in the late 1950s in connection with multi-player games (in the von Neumann–Morgenstern sense), is still open.

During the period when Cauchy worked on finite groups, he proved that a primitive group of degree prime plus one is 2-transitive. This was only refuted in the 1960s, when Charles Sims, Peter Neumann and James Wiegold observed that, if S is a simple group, then S×S, acting on S (so that one factor acts on the left and the other on the right) is primitive but not 2-transitive, and that many simple groups, starting with A₅, have order a prime plus one. I wonder whether this construction gives infinitely many counterexamples to Cauchy’s theorem. This is a question in number theory rather than group theory: the only thing it takes from group theory is the list of orders of simple groups.

A related question has arisen in connection with semigroup theory. Let G be a transitive permutation group of degree n on Ω, and f a function from Ω to itself whose image has size k < n. The question asks: For which groups is it the case that the semigroup ⟨G,f⟩∖G is idempotent-generated? It is known that this is true for k = 2 if and only if G has the road closure property, which asserts the following: if O is an orbit of G on the set of 2-subsets of Ω, and B a proper block of imprimitivity for G on O, then the graph with vertex set Ω and edge set O∖B is connected. The problem is: which primitive groups do (or do not) satisfy this property? This is a hard question about permutation groups, and while a lot is known, there is as yet no classification of these permutation groups.

References

[1] R. A. Bailey, P. J. Cameron, C. E. Praeger and Cs. Schneider, The geometry of diagonal groups, Trans. Amer. Math. Soc. 375 (2022), 5259–5311.

[2] P. J. Cameron, Transitivity of permutation groups on unordered sets, Math. Z. 148 (1976), 127–139.

[3] P. J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981), 1–22.

[4] P. J. Cameron, C. E. Praeger, J. Saxl and G. M. Seitz, On the Sims conjecture and distance-transitive graphs, Bull. London Math. Soc. 15 (1983), 499–506.

[5] P. J. Cameron and A. Rudvalis, A design and a geometry for the group F₂₂, Designs, Codes, Crypt. 44 (2007), 11–14.

[6] C. Jordan, Traité des substitutions et des équations algébriques, Gauthier-Villars, Paris, 1870.

[7] L. L. Scott, Representations in characteristic p, The Santa Cruz Conference on Finite Groups, Proc. Symp. Pure Math. 37, Amer. Math. Soc. 1981, pp. 319–331.