Association schemes with given stratum dimensions: on a paper of Peter M. Neumann

Anagnostopoulou-Merkouri, Marina¹ ; Cameron, Peter J.²

¹ School of Mathematics University of Bristol Bristol BS8 1QU UK
² School of Mathematics and Statistics University of St Andrews St Andrews Fife KY16 9SS UK

Algebraic Combinatorics, Tome 6 (2023) no. 5, pp. 1189-1210

Résumé

In January 1969, Peter M. Neumann wrote a paper entitled “Primitive permutation groups of degree $3p$”. The main theorem placed restrictions on the parameters of a primitive but not $2$-transitive permutation group of degree three times a prime. The paper was never published, and the results have been superseded by stronger theorems depending on the classification of the finite simple groups, for example a classification of primitive groups of odd degree.

However, there are further reasons for being interested in this paper. First, it was written at a time when combinatorial techniques were being introduced into the theory of finite permutation groups, and the paper gives a very good summary and application of these techniques. Second, like its predecessor by Helmut Wielandt on primitive groups of degree $2p$, it can be re-interpreted as a combinatorial result concerning association schemes whose common eigenspaces have dimensions of a rather limited form. This result uses neither the primality of $p$ nor the existence of a permutation group related to the combinatorial structure. We extract these results and give details of the related combinatorics.

Détail

Reçu le : 2022-09-07
Révisé le : 2023-01-17
Accepté le : 2023-03-26
Publié le : 2023-11-07

DOI : 10.5802/alco.307

Classification : 05E30, 20B15
Keywords: association scheme, permutation group, strongly regular graph

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits