A $(v,k,\lambda , \mu )$-partial difference set (PDS) is a subset $D$ of a group $G$ such that $|G| = v$, $|D| = k$, and every nonidentity element $x$ of $G$ can be written in either $\lambda $ or $\mu $ different ways as a product $gh^{-1}$, depending on whether or not $x$ is in $D$. Assuming the identity is not in $D$ and $D$ is inverse-closed, the corresponding Cayley graph $\mathrm{Cay}(G,D)$ will be strongly regular. Partial difference sets have been the subject of significant study, especially in abelian groups, but relatively little is known about PDSs in nonabelian groups. While many techniques useful for abelian groups fail to translate to a nonabelian setting, the purpose of this paper is to show that examples and constructions using abelian groups can be modified to generate several examples in nonabelian groups. In particular, in this paper we use such techniques to construct the first known examples of PDSs in nonabelian groups of order $q^{2m}$, where $q$ is a power of an odd prime $p$ and $m \ge 2$. The groups constructed can have exponent as small as $p$ or as large as $p^r$ in a group of order $p^{2r}$. Furthermore, we construct what we believe are the first known Paley-type PDSs in nonabelian groups and what we believe are the first examples of Paley–Hadamard difference sets in nonabelian groups, and, using analogues of product theorems for abelian groups, we obtain several examples of each. We conclude the paper with several possible future research directions.
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Keywords: partial difference set, difference set
Davis, James A. 1; Polhill, John 2; Smith, Ken 3; Swartz, Eric 4
CC-BY 4.0
@article{ALCO_2025__8_2_399_0,
author = {Davis, James A. and Polhill, John and Smith, Ken and Swartz, Eric},
title = {Nonabelian partial difference sets constructed using abelian techniques},
journal = {Algebraic Combinatorics},
pages = {399--419},
publisher = {The Combinatorics Consortium},
volume = {8},
number = {2},
year = {2025},
doi = {10.5802/alco.416},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.416/}
}
TY - JOUR AU - Davis, James A. AU - Polhill, John AU - Smith, Ken AU - Swartz, Eric TI - Nonabelian partial difference sets constructed using abelian techniques JO - Algebraic Combinatorics PY - 2025 SP - 399 EP - 419 VL - 8 IS - 2 PB - The Combinatorics Consortium UR - https://alco.centre-mersenne.org/articles/10.5802/alco.416/ DO - 10.5802/alco.416 LA - en ID - ALCO_2025__8_2_399_0 ER -
%0 Journal Article %A Davis, James A. %A Polhill, John %A Smith, Ken %A Swartz, Eric %T Nonabelian partial difference sets constructed using abelian techniques %J Algebraic Combinatorics %D 2025 %P 399-419 %V 8 %N 2 %I The Combinatorics Consortium %U https://alco.centre-mersenne.org/articles/10.5802/alco.416/ %R 10.5802/alco.416 %G en %F ALCO_2025__8_2_399_0
Davis, James A.; Polhill, John; Smith, Ken; Swartz, Eric. Nonabelian partial difference sets constructed using abelian techniques. Algebraic Combinatorics, Volume 8 (2025) no. 2, pp. 399-419. doi : 10.5802/alco.416. https://alco.centre-mersenne.org/articles/10.5802/alco.416/
[1] Amorphic association schemes from bent partitions, Discrete Math., Volume 347 (2024) no. 1, Paper no. 113658, 13 pages | DOI | MR | Zbl
[2] Finite geometry and combinatorial applications, London Mathematical Society Student Texts, 82, Cambridge University Press, Cambridge, 2015, xii+285 pages | DOI | MR
[3] Negative Latin square type partial difference sets in nonabelian groups of order 64, Finite Fields Appl., Volume 81 (2022), Paper no. 102044, 11 pages | DOI | MR | Zbl
[4] The maximal subgroups of the low-dimensional finite classical groups, London Mathematical Society Lecture Note Series, 407, Cambridge University Press, Cambridge, 2013, xiv+438 pages (With a foreword by Martin Liebeck) | DOI | MR
[5] Strongly regular graphs, Encyclopedia of Mathematics and its Applications, 182, Cambridge University Press, Cambridge, 2022, xvii+462 pages | DOI | MR
[6] Partial difference sets in -groups, Arch. Math. (Basel), Volume 63 (1994) no. 2, pp. 103-110 | DOI | MR | Zbl
[7] Negative Latin square type partial difference sets in nonelementary abelian 2-groups, J. London Math. Soc. (2), Volume 70 (2004) no. 1, pp. 125-141 | DOI | MR | Zbl
[8] Automorphisms of strongly regular graphs with applications to partial difference sets, Des. Codes Cryptogr., Volume 79 (2016) no. 3, pp. 471-485 | DOI | MR | Zbl
[9] Partial difference sets and amorphic Cayley schemes in non-abelian 2-groups, J. Combin. Des., Volume 28 (2020) no. 4, pp. 273-293 | DOI | MR | Zbl
[10] Constructions of strongly regular Cayley graphs and skew Hadamard difference sets from cyclotomic classes, Combinatorica, Volume 35 (2015) no. 4, pp. 413-434 | DOI | MR | Zbl
[11] GAP – Groups, Algorithms, and Programming, Version 4.14.0 (2024) https://www.gap-system.org
[12] Switching of edges in strongly regular graphs. I. A family of partial difference sets on 100 vertices, Electron. J. Combin., Volume 10 (2003), Paper no. 17, 31 pages | DOI | MR | Zbl
[13] The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, 129, Cambridge University Press, Cambridge, 1990, x+303 pages | DOI | MR
[14] The axiomatics of cellular rings, Investigations in the algebraic theory of combinatorial objects (Russian), Vsesoyuz. Nauchno-Issled. Inst. Sistem. Issled., Moscow, 1985, pp. 6-32 | MR | Zbl
[15] Constructions of partial difference sets and relative difference sets on -groups, Bull. London Math. Soc., Volume 22 (1990) no. 6, pp. 533-539 | DOI | MR | Zbl
[16] Partial difference sets with Paley parameters, Bull. London Math. Soc., Volume 27 (1995) no. 6, pp. 553-564 | DOI | MR | Zbl
[17] A survey of partial difference sets, Des. Codes Cryptogr., Volume 4 (1994) no. 3, pp. 221-261 | DOI | MR | Zbl
[18] Partial difference sets in , Discrete Math., Volume 343 (2020) no. 4, Paper no. 111744, 22 pages | DOI | MR | Zbl
[19] Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets, Finite Fields Appl., Volume 50 (2018), pp. 222-250 | DOI | MR | Zbl
[20] On generalized quadrangles with a group of automorphisms acting regularly on the point set, difference sets with as multiplier and a conjecture of Ghinelli, European J. Combin., Volume 110 (2023), Paper no. 103681, 10 pages | DOI | MR | Zbl
[21] On Orthogonal Matrices, J. Math. Phys., Volume 12 (1933) no. 1–4, pp. 311-320 | DOI | Zbl
[22] New negative Latin square type partial difference sets in nonelementary abelian 2-groups and 3-groups, Des. Codes Cryptogr., Volume 46 (2008) no. 3, pp. 365-377 | DOI | MR | Zbl
[23] Negative Latin square type partial difference sets and amorphic association schemes with Galois rings, J. Combin. Des., Volume 17 (2009) no. 3, pp. 266-282 | DOI | MR | Zbl
[24] Paley type partial difference sets in non -groups, Des. Codes Cryptogr., Volume 52 (2009) no. 2, pp. 163-169 | DOI | MR | Zbl
[25] A new family of partial difference sets in 3-groups, Des. Codes Cryptogr., Volume 87 (2019) no. 7, pp. 1639-1646 | DOI | MR | Zbl
[26] Genuinely nonabelian partial difference sets, J. Combin. Des., Volume 32 (2024) no. 7, pp. 351-370 | DOI | MR | Zbl
[27] Constructions of nested partial difference sets with Galois rings, Des. Codes Cryptogr., Volume 25 (2002) no. 3, pp. 299-309 | DOI | MR | Zbl
[28] Non-abelian Hadamard difference sets, J. Combin. Theory Ser. A, Volume 70 (1995) no. 1, pp. 144-156 | DOI | MR | Zbl
[29] A family of difference sets, Canadian J. Math., Volume 10 (1958), pp. 73-77 | DOI | MR | Zbl
[30] A construction of a partial difference set in the extraspecial groups of order with exponent , Des. Codes Cryptogr., Volume 75 (2015) no. 2, pp. 237-242 | DOI | MR | Zbl
[31] Restrictions on parameters of partial difference sets in nonabelian groups, J. Combin. Des., Volume 29 (2021) no. 1, pp. 38-51 | DOI | MR | Zbl
[32] Some implications on amorphic association schemes, J. Combin. Theory Ser. A, Volume 117 (2010) no. 2, pp. 111-127 | DOI | MR | Zbl
[33] Strongly regular decompositions of the complete graph, J. Algebraic Combin., Volume 17 (2003) no. 2, pp. 181-201 | DOI | MR | Zbl
[34] The finite simple groups, Graduate Texts in Mathematics, 251, Springer-Verlag London, Ltd., London, 2009, xvi+298 pages | DOI | MR
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