## Peter Neumann’s 3p paper

In 1955, Helmut Wielandt published a paper proving the following theorem:

Let G be a primitive permutation group of degree 2p, where p is a prime greater than 3, which is not doubly transitive. Then p = 2a2+2a+1 for some positive integer a, and G has rank 3 and subdegrees a(2a+1) and (a+1)(2a+1).

The proof involved first showing that the permutation character decomposes into three irreducible constituents, the principal character and characters of degree p−1 and p, followed by analysis of what we would now call the coherent configuration associated with G. Wielandt was a student of Schur, and had essentially stripped away the algebraic part of the theory of Schur rings to leave a purely combinatorial object.

Indeed, the second part of Wielandt’s argument can be rewritten to show the following result (I am not sure who originally proved this, but you can find it as Theorem (2.20) in my book with Jack van Lint):

Let Γ be a strongly regular graph on 2n vertices whose eigenvalues have multiplicities 1, n−1 and n. Then, up to complementation, either Γ consists of n pairwise disjoint edges, or n = 2a2+2a+1 for some positive integer a, and Γ has valency a(2a+1).

This result is much more general; the only examples satisfying Wielandt’s hypothesis are the symmetric and alternating groups of degree 5 acting on the ten unordered pairs from 5 (we need the Classification of Finite Simple Groups to show there are no more, a tool which of course was not available to Wielandt), whereas there are many strongly regular graphs satisfying the hypotheses (after the Petersen graph of degree 10 there are several of degree 26, etc.)

But I am getting out of historical order here.

When I arrived in Oxford as a doctoral student in 1968, Peter Neumann gave me Wielandt’s book Finite Permutation Groups to read, and then gave me a copy of a typescript over 50 pages long he had just finished, proving an analogue of Wielandt’s theorem for primitive groups of degree 3p.

In the time since Wielandt’s paper, Walter Feit had proved a theorem about degrees of linear groups which greatly reduced the work required to decompose the permutation character. But the second combinatorial part is much more difficult, since there are many possible decompositions, including some in which a character has multiplicity greater than 1, and very detailed analysis is required in each case. The final result has not a single quadratic expression for p, but three different expressions, together with three exceptional values.

Leonard Scott, who has a role in this story (as he had proved similar results), kindly sent me a scan of the typescript, mine having been lost many years ago, together with scans of various items showing that he and Peter had planned a collaboration on this material which never came about. The paper remained unpublished and virtually inaccessible.

Until now.

I have just posted on the arXiv a version of the paper re-typed in LaTeX: it is 2204.06926.

This is motivated in part by the fact that this paper was written at the time when the combinatorial methods in permutation groups developed by Wielandt and Donald Higman were very similar to those used by statisticians (who, following R. C. Bose, called them association schemes) and mathematicians/computer scientists led by Boris Weisfeiler in the former Soviet Union working on graph isomorphism (who called them cellular algebras). Peter’s paper gives a very nice exposition of this theory.

The other reason is that St Andrews undergraduate Marina Anagnostopoulou-Merkouri is working on extracting purely combinatorial information from the paper, similar to the result on strongly regular graphs which derives from Wielandt’s arguments, and needs to be able to refer to the paper.

It should be said that the actual theorem proved in the paper is, like Wielandt’s, superseded by consequences of the Classification of Finite Simple Groups. For example, in the 1980s, Bill Kantor, and independently Martin Liebeck and Jan Saxl, found (in some sense) all the primitive permutation groups of odd degree, and their results show that the short list of examples in Peter Neumann’s paper is complete.