TY - JOUR

T1 - No sporadic almost simple group acts primitively on the points of a generalised quadrangle

AU - Bamberg, John

AU - Evans, James

N1 - DBLP License: DBLP's bibliographic metadata records provided through http://dblp.org/ are distributed under a Creative Commons CC0 1.0 Universal Public Domain Dedication. Although the bibliographic metadata records are provided consistent with CC0 1.0 Dedication, the content described by the metadata records is not. Content may be subject to copyright, rights of privacy, rights of publicity and other restrictions.

PY - 2021/4

Y1 - 2021/4

N2 - A generalised quadrangle is a point–line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given a line L and a point p not incident with L, there is a unique point on L collinear with p. They are a specific case of the generalised polygons introduced by Tits (1959), and these structures and their automorphism groups are of some importance in finite geometry. An integral part of understanding the automorphism groups of finite generalised quadrangles is knowing which groups can act primitively on their points, and in particular, which almost simple groups arise as automorphism groups. We show that no sporadic almost simple group can act primitively on the points of a finite (thick) generalised quadrangle. We also present two new ideas contributing towards analysing point-primitive groups acting on generalised quadrangles. The first is the outline and implementation of an algorithm for determining whether a given group can act primitively on the points of some generalised quadrangle. The second is the discussion of a conjecture resulting from observations made in the course of this work: any group acting primitively on the points of a generalised quadrangle must either act transitively on lines or have exactly two line-orbits, each containing half of the lines.

AB - A generalised quadrangle is a point–line incidence geometry G such that: (i) any two points lie on at most one line, and (ii) given a line L and a point p not incident with L, there is a unique point on L collinear with p. They are a specific case of the generalised polygons introduced by Tits (1959), and these structures and their automorphism groups are of some importance in finite geometry. An integral part of understanding the automorphism groups of finite generalised quadrangles is knowing which groups can act primitively on their points, and in particular, which almost simple groups arise as automorphism groups. We show that no sporadic almost simple group can act primitively on the points of a finite (thick) generalised quadrangle. We also present two new ideas contributing towards analysing point-primitive groups acting on generalised quadrangles. The first is the outline and implementation of an algorithm for determining whether a given group can act primitively on the points of some generalised quadrangle. The second is the discussion of a conjecture resulting from observations made in the course of this work: any group acting primitively on the points of a generalised quadrangle must either act transitively on lines or have exactly two line-orbits, each containing half of the lines.

KW - Generalised quadrangles

KW - Hemisystems

KW - m-ovoids

KW - Primitive permutation groups

KW - Sporadic groups

UR - http://www.scopus.com/inward/record.url?scp=85099835665&partnerID=8YFLogxK

UR - https://arxiv.org/abs/2007.06161

U2 - 10.1016/j.disc.2021.112291

DO - 10.1016/j.disc.2021.112291

M3 - Article

AN - SCOPUS:85099835665

SN - 0012-365X

VL - 344

SP - 112291

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 4

M1 - 4

ER -