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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: [email protected], [email protected]
FRANCIS HUNT
Affiliation:
Department of Mathematics, University of South Wales, Treforest, CF37 1DL, U.K. e-mails: [email protected], [email protected]
PABLO SPIGA
Affiliation:
Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca, Via Cozzi 55, 20125 Milano, Italy. e-mail: [email protected]
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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

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