ALGEBRAIC COMBINATORICS

no. 5
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, Volume 6 (2023) no. 5, pp. 1189-1210.

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 The paper was never published, and the results have been superseded by stronger theorems depending on the classification of the finite simple groups, for example a classification of primitive groups of odd degree.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

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.307
Classification: 05E30, 20B15
Keywords: association scheme, permutation group, strongly regular graph
Anagnostopoulou-Merkouri, Marina 1; 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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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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/

[1] Cameron, P. J.; van Lint, J. H. Designs, Graphs, Codes and their Links, London Mathematical Society Student Texts, 22, Cambridge University Press, Cambridge, 1991 | DOI

[2] Coolsaet, K.; Degraer, J. A computer assisted proof of the uniqueness of the Perkel graph, Design Code Cryptogr, Volume 34 (2005), pp. 155-171 | DOI | MR | Zbl

[3] Feit, W. On finite linear groups, J ALgebra, Volume 5 (1967), pp. 378-400 | DOI | MR | Zbl

[4] Gantmacher, Felix The Theory of Matrices, 2, AMS Chelsea Publishing, New York, 1998 (reprint of 1959 edition)

[5] GAP – Groups, Algorithms, and Programming, Version 4.12.2 (2022) https://www.gap-system.org

[6] Hanaki, Akihide; Miyamoto, Izumi Classification of association schemes of small vertices http://math.shinshu-u.ac.jp/~hanaki/as/ (visited on 3 July 2022)

[7] Kantor, William M. Primitive permutation groups of odd degree, and an application to finite projective planes, J. Algebra, Volume 106 (1987) no. 1, pp. 15-45 | DOI | MR | Zbl

[8] Kantor, William M.; Liebler, Robert A. The rank 3 permutation representations of the finite classical groups, Trans. Amer. Math. Soc., Volume 271 (1982) no. 1, pp. 1-71 | DOI | MR | Zbl

[9] Kaski, Petteri; Östergård, Patric R. J. The Steiner triple systems of order 19, Math. Comp., Volume 73 (2004) no. 248, pp. 2075-2092 | DOI | MR | Zbl

[10] Liebeck, Martin W. The affine permutation groups of rank three, Proc. London Math. Soc. (3), Volume 54 (1987) no. 3, pp. 477-516 | DOI | MR | Zbl

[11] Liebeck, Martin W.; Saxl, Jan The primitive permutation groups of odd degree, J. London Math. Soc. (2), Volume 31 (1985) no. 2, pp. 250-264 | DOI | MR | Zbl

[12] Liebeck, Martin W.; Saxl, Jan The finite primitive permutation groups of rank three, Bull. London Math. Soc., Volume 18 (1986) no. 2, pp. 165-172 | DOI | MR | Zbl

[13] Neumann, Peter M. Primitive permutation groups of degree 3p (1969) (unpublished manuscript)

[14] Neumann, Peter M. Primitive permutation groups of degree 3p, 2022 | arXiv

[15] Perkel, Manley Bounding the valency of polygonal graphs with odd girth, Canadian J. Math., Volume 31 (1979) no. 6, pp. 1307-1321 | DOI | MR | Zbl

[16] Reid, K. B.; Brown, Ezra Doubly regular tournaments are equivalent to skew Hadamard matrices, J. Combinatorial Theory Ser. A, Volume 12 (1972), pp. 332-338 | DOI | MR | Zbl

[17] Scott, L. L. Personal communication, 2022

[18] Wielandt, Helmut Primitive Permutationsgruppen vom Grad 2p, Math. Z., Volume 63 (1956), pp. 478-485 | DOI | MR | Zbl

[19] Wielandt, Helmut Finite permutation groups, Academic Press, New York-London, 1964, x+114 pages (translated from the German by R. Bercov)

[20] Wilbrink, H. A.; Brouwer, A. E. A (57,14,1) strongly regular graph does not exist, Nederl. Akad. Wetensch. Indag. Math., Volume 45 (1983) no. 1, pp. 117-121 | DOI | MR | Zbl

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