Ernie Shult

Ernie Shult died last month. I just heard the news, from Corneliu Hoffman’s mailing list.

I first met Ernie at a “microconference” on permutation groups organised in Oxford by Peter Neumann in the early 1970s when I was just starting out. Many of the gods of finite group theory were there: Shult, Leonard Scott, Charles Sims, I don’t remember the whole list. Ernie was one of the nicest people I knew, and I will end with a story from that meeting.

There was one of Shult’s theorems that I knew well at the time, the Graph Extension Theorem. This gave a sufficient condition for the automorphism group of a vertex-transitive graph to be the stabiliser of a point in a doubly transitive group. I would like to remember Ernie by tracing some of the secret history of this theorem.

Shult himself applied it to give a characterisation of the symplectic and orthogonal groups over GF(2). This led, via a generalisation by Jaap Seidel, to the very influential Buekenhout–Shult Theorem, which gave a simple test to recognise the geometries of all the classical groups, and was used for this in the Classification of Finite Simple Groups. But I want to trace a different thread.

The Graph Extension Theorem was closely related to Graham Higman’s concept of a regular two-graph, a collection of triples satisfying a cocycle condition. To explain the connection in highbrow language: because of vanishing cohomology, a cocyle is necessarily a coboundary, in this case of a graph, which is the graph in Shult’s theorem.

A little later, John McDermott came to Oxford to give a seminar. He, like me, began as a finite group theorist; he was talking about infinite permutation groups, and specifically highly homogeneous groups, groups which act transitively on the set of k-element subsets of their domain for all k, which fail to be k-transitive (transitive on ordered k-tuples) for some k. The standard examples are linear orders such as the rational or real numbers.

John saw that you could obtain a larger group by preserving or reversing the linear order: this group is 2-transitive (but not 3-transitive). The main part of his talk was the construction of further examples of 2-transitive groups. He showed that linear orders like the rational or real numbers satisfy the hypotheses of an “oriented” version of Shult’s Graph Extension Theorem (which he proved), and thus are point stabilisers in 2-transitive groups.

Shult’s original construction is related to Higman’s regular two-graphs, which are sets of triples (or, said otherwise, relations which say “yes” or “no” to every unordered triple). Analogously, McDermott’s construction should be related to “oriented triples”, which give a cyclic orientation to every unordered triple. The obvious gadget which does this is a circular order. So it turned out that McDermott had constructed the circular order obtained from a linear order.

Once I noticed this, it was simple to observe that you could preserve or reverse the circular order (obtaining a 3-transitive group), and I was able to prove that these were all the possibilities.

Later, when I learned about the random graph, I realised that it too satisfied Shult’s hypotheses (in their original form), and so its automorphism group has a transitive extension. This led to Simon Thomas’ classification of the reducts of the random graph.

A fine body of theory to have grown out of quite a short paper (which, by the way, was published in a conference proceedings – research assessors please note!)

Back to the miniconference. At a certain point there was a party in my house. The only thing I remember from this party is that at a certain point, Ernie Shult and Leonard Scott were discussing me. Ernie said that I was a very gentle person, but Leonard disagreed, saying that I had a core of toughness (I don’t remember the precise word he used). It is not for me to judge who was right, but how characteristic of Ernie to see the good in someone.