ALGEBRAIC COMBINATORICS

no. 4
Analysing flag-transitive point-imprimitive 2-designs
Devillers, Alice1; Praeger, Cheryl E.1
1 Centre for the Mathematics of Symmetry and Computation University of Western Australia Perth, WA 6009, Australia.
Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 1041-1055.

In this paper we develop several general methods for analysing flag-transitive point-imprimitive 2-designs, which give restrictions on both the automorphisms and parameters of such designs. These constitute a tool-kit for analysing these designs and their groups. We apply these methods to complete the classification of flag-transitive, point-imprimitive 2-(v,k,λ) designs with λ at most 4.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.297
Classification: 05B05, 20B25
Keywords: block design, 2-design, flag-transitive group, imprimitive group
Devillers, Alice 1; Praeger, Cheryl E. 1

1 Centre for the Mathematics of Symmetry and Computation University of Western Australia Perth, WA 6009, Australia.
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{ALCO_2023__6_4_1041_0,
     author = {Devillers, Alice and Praeger, Cheryl E.},
     title = {Analysing flag-transitive point-imprimitive $2$-designs},
     journal = {Algebraic Combinatorics},
     pages = {1041--1055},
     publisher = {The Combinatorics Consortium},
     volume = {6},
     number = {4},
     year = {2023},
     doi = {10.5802/alco.297},
     language = {en},
     url = {https://alco.centre-mersenne.org/articles/10.5802/alco.297/}
}
TY  - JOUR
AU  - Devillers, Alice
AU  - Praeger, Cheryl E.
TI  - Analysing flag-transitive point-imprimitive $2$-designs
JO  - Algebraic Combinatorics
PY  - 2023
SP  - 1041
EP  - 1055
VL  - 6
IS  - 4
PB  - The Combinatorics Consortium
UR  - https://alco.centre-mersenne.org/articles/10.5802/alco.297/
DO  - 10.5802/alco.297
LA  - en
ID  - ALCO_2023__6_4_1041_0
ER  - 
%0 Journal Article
%A Devillers, Alice
%A Praeger, Cheryl E.
%T Analysing flag-transitive point-imprimitive $2$-designs
%J Algebraic Combinatorics
%D 2023
%P 1041-1055
%V 6
%N 4
%I The Combinatorics Consortium
%U https://alco.centre-mersenne.org/articles/10.5802/alco.297/
%R 10.5802/alco.297
%G en
%F ALCO_2023__6_4_1041_0
Devillers, Alice; Praeger, Cheryl E. Analysing flag-transitive point-imprimitive $2$-designs. Algebraic Combinatorics, Volume 6 (2023) no. 4, pp. 1041-1055. doi : 10.5802/alco.297. https://alco.centre-mersenne.org/articles/10.5802/alco.297/

[1] Bosma, Wieb; Cannon, John; Playoust, Catherine The Magma algebra system. I. The user language, J. Symbolic Comput., Volume 24 (1997) no. 3-4, pp. 235-265 Computational algebra and number theory (London, 1993) | DOI | MR | Zbl

[2] Cameron, Peter J.; Praeger, Cheryl E. Constructing flag-transitive, point-imprimitive designs, J. Algebraic Combin., Volume 43 (2016) no. 4, pp. 755-769 | DOI | MR | Zbl

[3] Coutts, Hannah J.; Quick, Martyn; Roney-Dougal, Colva M. The primitive permutation groups of degree less than 4096, Comm. Algebra, Volume 39 (2011) no. 10, pp. 3526-3546 (Implemented into Magma) | DOI | MR | Zbl

[4] Davies, Huw Flag-transitivity and primitivity, Discrete Math., Volume 63 (1987) no. 1, pp. 91-93 | DOI | MR | Zbl

[5] Dembowski, Peter Finite geometries, Classics in Mathematics, Springer-Verlag, Berlin, 1997, xii+375 pages (Reprint of the 1968 original) | MR

[6] Devillers, Alice; Liang, Hongxue; Praeger, Cheryl E.; Xia, Binzhou On flag-transitive 2-(v,k,2) designs, J. Combin. Theory Ser. A, Volume 177 (2021), Paper no. 105309, 45 pages | DOI | MR | Zbl

[7] Devillers, Alice; Praeger, Cheryl E. On flag-transitive imprimitive 2-designs, J. Combin. Des., Volume 29 (2021) no. 8, pp. 552-574 | DOI | MR | Zbl

[8] Dixon, John D.; Mortimer, Brian Permutation groups, Graduate Texts in Mathematics, 163, Springer-Verlag, New York, 1996, xii+346 pages | DOI | MR

[9] Higman, D. G.; McLaughlin, J. E. Geometric ABA-groups, Illinois J. Math., Volume 5 (1961), pp. 382-397 | DOI | MR | Zbl

[10] Hughes, D. R.; Piper, F. C. Design theory, Cambridge University Press, Cambridge, 1985, viii+240 pages | DOI | MR

[11] Law, Maska; Praeger, Cheryl E.; Reichard, Sven Flag-transitive symmetric 2-(96,20,4)-designs, J. Combin. Theory Ser. A, Volume 116 (2009) no. 5, pp. 1009-1022 | DOI | MR | Zbl

[12] Mandić, Joško; Šubašić, Aljoša Flag-transitive and point-imprimitive symmetric designs with λ≤10, J. Combin. Theory Ser. A, Volume 189 (2022), Paper no. 105620, 21 pages | DOI | MR | Zbl

[13] Montinaro, Alessandro Flag-transitive, point-imprimitive symmetric 2-(v,k,λ) designs with k>λ(λ-3)/2 (2022) | arXiv

[14] Montinaro, Alessandro The symmetric 2-(v,k,λ) designs, with k>λ(λ-3)/2, admitting a flag-transitive, point-imprimitive automorphism group are known. (2022) | arXiv

[15] Praeger, Cheryl E. The flag-transitive symmetric designs with 45 points, blocks of size 12, and 3 blocks on every point pair, Des. Codes Cryptogr., Volume 44 (2007) no. 1-3, pp. 115-132 | DOI | MR | Zbl

[16] Praeger, Cheryl E.; Schneider, Csaba Permutation groups and Cartesian decompositions, London Mathematical Society Lecture Note Series, 449, Cambridge University Press, Cambridge, 2018, xiii+323 pages | DOI | MR

[17] Praeger, Cheryl E.; Zhou, Shenglin Imprimitive flag-transitive symmetric designs, J. Combin. Theory Ser. A, Volume 113 (2006) no. 7, pp. 1381-1395 | DOI | MR | Zbl

[18] Regueiro, Eugenia O’Reilly On primitivity and reduction for flag-transitive symmetric designs, J. Combin. Theory Ser. A, Volume 109 (2005) no. 1, pp. 135-148 | DOI | MR | Zbl

[19] Zhao, Yanwei; Zhou, Shenglin Flag-transitive 2-(v,k,λ) designs with r>λ(k-3), Des. Codes Cryptogr., Volume 90 (2022) no. 4, pp. 863-869 | DOI | MR | Zbl

Cited by Sources: