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A classification of finite primitive IBIS groups with alternating socle

  • Melissa Lee and Pablo Spiga ORCID logo EMAIL logo
Published/Copyright: February 23, 2023

Abstract

Let 𝐺 be a finite permutation group on Ω. An ordered sequence ( ω 1 , , ω ) of elements of Ω is an irredundant base for 𝐺 if the pointwise stabilizer is trivial and no point is fixed by the stabilizer of its predecessors. If all irredundant bases of 𝐺 have the same cardinality, 𝐺 is said to be an IBIS group. Lucchini, Morigi and Moscatiello have proved a theorem reducing the problem of classifying finite primitive IBIS groups 𝐺 to the case that the socle of 𝐺 is either abelian or non-abelian simple. In this paper, we classify the finite primitive IBIS groups having socle an alternating group. Moreover, we propose a conjecture aiming to give a classification of all almost simple primitive IBIS groups.

Award Identifier / Grant number: DE230100579

Funding statement: The first author acknowledges the support of an Australian Research Council Discovery Early Career Researcher Award (project number DE230100579).

Acknowledgements

The authors also thank the referee for their careful reading and helpful suggestions.

  1. Communicated by: Timothy C. Burness

References

[1] R. F. Bailey, Uncoverings-by-bases for base-transitive permutation groups, Des. Codes Cryptogr. 41 (2006), no. 2, 153–176. 10.1007/s10623-006-9005-xSearch in Google Scholar

[2] T. C. Burness, On base sizes for actions of finite classical groups, J. Lond. Math. Soc. (2) 75 (2007), no. 3, 545–562. 10.1112/jlms/jdm033Search in Google Scholar

[3] T. C. Burness, R. M. Guralnick and J. Saxl, On base sizes for symmetric groups, Bull. Lond. Math. Soc. 43 (2011), no. 2, 386–391. 10.1112/blms/bdq123Search in Google Scholar

[4] T. C. Burness, M. W. Liebeck and A. Shalev, Base sizes for simple groups and a conjecture of Cameron, Proc. Lond. Math. Soc. (3) 98 (2009), no. 1, 116–162. 10.1112/plms/pdn024Search in Google Scholar

[5] T. C. Burness, E. A. O’Brien and R. A. Wilson, Base sizes for sporadic simple groups, Israel J. Math. 177 (2010), 307–333. 10.1007/s11856-010-0048-3Search in Google Scholar

[6] J. Cáceres, D. Garijo, A. González, A. Márquez and M. L. Puertas, The determining number of Kneser graphs, Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 1–14. 10.46298/dmtcs.634Search in Google Scholar

[7] P. J. Cameron, Permutation Groups, London Math. Soc. Stud. Texts 45, Cambridge University, Cambridge, 1999. Search in Google Scholar

[8] P. J. Cameron and D. G. Fon-Der-Flaass, Bases for permutation groups and matroids, European J. Combin. 16 (1995), no. 6, 537–544. 10.1016/0195-6698(95)90035-7Search in Google Scholar

[9] N. Gill, B. Lodà and P. Spiga, On the height and relational complexity of a finite permutation group, Nagoya Math. J. 246 (2022), 372–411. 10.1017/nmj.2021.6Search in Google Scholar

[10] Z. Halasi, On the base size for the symmetric group acting on subsets, Studia Sci. Math. Hungar. 49 (2012), no. 4, 492–500. 10.1556/sscmath.49.2012.4.1222Search in Google Scholar

[11] A. Lucchini, M. Morigi and M. Moscatiello, Primitive permutation IBIS groups, J. Combin. Theory Ser. A 184 (2021), Paper No. 105516. 10.1016/j.jcta.2021.105516Search in Google Scholar

[12] T. Maund, Bases for permutation groups, D. Phil. thesis, Oxford University, 1989. Search in Google Scholar

[13] J. Morris and P. Spiga, On the base size of the symmetric and the alternating group acting on partitions, J. Algebra 587 (2021), 569–593. 10.1016/j.jalgebra.2021.08.009Search in Google Scholar

[14] M. Moscatiello and C. M. Roney-Dougal, Base sizes of primitive permutation groups, Monatsh. Math. 198 (2022), no. 2, 411–443. 10.1007/s00605-021-01599-5Search in Google Scholar

[15] M. Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145. 10.2307/1970423Search in Google Scholar

[16] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.9.3, 2018. Search in Google Scholar

Received: 2022-06-03
Revised: 2023-01-23
Published Online: 2023-02-23
Published in Print: 2023-09-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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