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Cherlin's conjecture for almost simple groups of Lie rank 1

Published online by Cambridge University Press:  05 July 2018

NICK GILL ,
FRANCIS HUNT  and
PABLO SPIGA
NICK GILL
Affiliation:
Department of Mathematics, University of South Wales, Treforest, CF37 1DL, U.K. e-mails: nick.gill@southwales.ac.uk, francis.hunt@southwales.ac.uk
FRANCIS HUNT
Affiliation:
Department of Mathematics, University of South Wales, Treforest, CF37 1DL, U.K. e-mails: nick.gill@southwales.ac.uk, francis.hunt@southwales.ac.uk
PABLO SPIGA
Affiliation:
Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy. e-mail: pablo.spiga@unimib.it
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Abstract

A permutation group G on a set Ω is said to be binary if, for every n ∈ ℕ and for every I, J ∈ Ωn, the n-tuples I and J are in the same G-orbit if and only if every pair of entries from I is in the same G-orbit to the corresponding pair from J. This notion arises from the investigation of the relational complexity of finite homogeneous structures.

Cherlin has conjectured that the only finite primitive binary permutation groups are the symmetric groups Sym(n) with their natural action, the groups of prime order, and the affine groups V ⋊ O(V) where V is a vector space endowed with an anisotropic quadratic form.

We prove Cherlin's conjecture, concerning binary primitive permutation groups, for those groups with socle isomorphic to PSL2(q), 2B2(q), 2G2(q) or PSU3(q). Our method uses the notion of a “strongly non-binary action”.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 3 , November 2019 , pp. 417 - 435
Copyright
Copyright © Cambridge Philosophical Society 2018 

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References

REFERENCES

[1] Bosma, W., Cannon, J. and Playoust, C. The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265.Google Scholar
[2] Bray, J. N., Holt, D. F. and Roney-Dougal, C. M. The Maximal Subgroups of the Low-Dimensional Finite Classical Groups (Cambridge University Press, 2013).Google Scholar
[3] Cherlin, G. Sporadic homogeneous structures. In The Gelfand Mathematical Seminars, 1996–1999. Dedicated to the memory of Chih-Han Sah (Boston, MA: Birkhäuser, 2000), pages 15–48.Google Scholar
[4] Cherlin, G. On the relational complexity of a finite permutation group. J. Algebraic Combin. 43 (2016), 339–374.Google Scholar
[5] Dalla Volta, F., Gill, N. and Spiga, P. Cherlin's conjecture for almost simple groups with socle a sporadic simple group. Pacific J. Mathematics, to appear.Google Scholar
[6] Dixon, J. D. and Mortimer, B. Permutation groups Graduate Texts in Math. vol. 163 (Springer-Verlag, New York, 1996).Google Scholar
[7] The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.8.4 (2016).Google Scholar
[8] Gill, N., Liebeck, M. and Spiga, P. Cherlin's conjecture for finite groups of Lie type. In preparation.Google Scholar
[9] Gill, N. and Spiga, P. Cherlin's conjecture: alternating and classical groups (2016). Preprint available on the Math arXiv: https://arxiv.org/abs/1610.01792.Google Scholar
[10] Gorenstein, D., Lyons, R. and Solomon, R. The classification of the finite simple groups. Number 3. Part I. Chapter A. Math. Surv. Monogr. vol 40 (American Mathematical Society, Providence, RI, 1998).Google Scholar
[11] Kleidman, P. B. The maximal subgroups of the Chevalley groups G 2(q) with q odd, the Ree groups 2G 2(q), and their automorphism groups. J. Algebra 117 (1988), 30–71.Google Scholar
[12] Liebeck, M. W. and Saxl, J. Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces. Proc. London Math. Soc. (3) 63 (1991), 266–314.Google Scholar
[13] Suzuki, M. On a class of doubly transitive groups. Ann. Math. (2) 75 (1962), 105–145.Google Scholar
[14] Wielandt, H. Finite Permutation Groups. Translated from the German by R. Bercov (Academic Press, New York-London, 1964).Google Scholar
[15] Wiscons, J. A reduction theorem for primitive binary permutation groups. Bull. Lond. Math. Soc. 48 (2016), 291–299.Google Scholar
[16] Wiscons, J. The relational complexity of some small primitive permutation groups. Available online: http://webpages.csus.edu/wiscons/research/RCompDataLatex.pdf.Google Scholar