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 The paper was never published, and the results have been superseded by stronger theorems depending on the classification of the finite simple groups, for example a classification of primitive groups of odd degree.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.

Revised:

Accepted:

Published online:

Keywords: association scheme, permutation group, strongly regular graph

^{1}; Cameron, Peter J.

^{2}

@article{ALCO_2023__6_5_1189_0, author = {Anagnostopoulou-Merkouri, Marina and Cameron, Peter J.}, title = {Association schemes with given stratum dimensions: on a paper of {Peter~M.~Neumann}}, journal = {Algebraic Combinatorics}, pages = {1189--1210}, publisher = {The Combinatorics Consortium}, volume = {6}, number = {5}, year = {2023}, doi = {10.5802/alco.307}, language = {en}, url = {https://alco.centre-mersenne.org/articles/10.5802/alco.307/} }

TY - JOUR AU - Anagnostopoulou-Merkouri, Marina AU - Cameron, Peter J. TI - Association schemes with given stratum dimensions: on a paper of Peter M. Neumann JO - Algebraic Combinatorics PY - 2023 SP - 1189 EP - 1210 VL - 6 IS - 5 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.307/ DO - 10.5802/alco.307 LA - en ID - ALCO_2023__6_5_1189_0 ER -

%0 Journal Article %A Anagnostopoulou-Merkouri, Marina %A Cameron, Peter J. %T Association schemes with given stratum dimensions: on a paper of Peter M. Neumann %J Algebraic Combinatorics %D 2023 %P 1189-1210 %V 6 %N 5 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.307/ %R 10.5802/alco.307 %G en %F ALCO_2023__6_5_1189_0

Anagnostopoulou-Merkouri, Marina; Cameron, Peter J. Association schemes with given stratum dimensions: on a paper of Peter M. Neumann. Algebraic Combinatorics, Volume 6 (2023) no. 5, pp. 1189-1210. doi : 10.5802/alco.307. https://alco.centre-mersenne.org/articles/10.5802/alco.307/

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