Primitive flag-transitive generalized quadrangles

Abstract

Let 𝒬 be a finite thick generalized quadrangle of order (s,t) with a flag-transitive automorphism group G. A well-known conjecture of Kantor from the early 1990s asserts that up to duality, 𝒬 is either of classical type, or it is one of two sporadic examples of order (3,5) or (15,17). Although this fundamental problem in finite geometry remains open, it has been resolved in some special cases. For example, a theorem of Buekenhout and Van Maldeghem from 1994 proves Kantor's conjecture under the stronger assumption that G acts distance-transitively on the set of points of 𝒬. In this paper, we study Kantor's conjecture in the setting where G acts primitively on both the points and lines of 𝒬. Here a theorem of Bamberg et al. implies that G is almost simple with socle X a finite simple group of Lie type, and several special cases where the Lie rank of X is at most $2$ have recently been settled in a sequence of papers. Our main theorem resolves the problem for all almost simple groups of Lie type, thereby completing the proof of Kantor's conjecture for generalized quadrangles with a flag-transitive automorphism group acting primitively on points and lines.

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BibTeX RIS

BibTeX

@misc{burness2026primitive,
  title = {{Primitive flag-transitive generalized quadrangles}},
  author = {Timothy C. Burness and Tao Feng and Jianbing Lu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38819},
  eprint = {2610.38819},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38819},
}

RIS

TY  - UNPB
TI  - Primitive flag-transitive generalized quadrangles
AU  - Timothy C. Burness
AU  - Tao Feng
AU  - Jianbing Lu
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38819
UR  - https://arxiv.org/abs/2610.38819
AB  - Let 𝒬 be a finite thick generalized quadrangle of order (s,t) with a flag-transitive automorphism group G. A well-known conjecture of Kantor from the early 1990s asserts that up to duality, 𝒬 is either of classical type, or it is one of two sporadic examples of order (3,5) or (15,17). Although this fundamental problem in finite geometry remains open, it has been resolved in some special cases. For example, a theorem of Buekenhout and Van Maldeghem from 1994 proves Kantor's conjecture under the stronger assumption that G acts distance-transitively on the set of points of 𝒬. In this paper, we study Kantor's conjecture in the setting where G acts primitively on both the points and lines of 𝒬. Here a theorem of Bamberg et al. implies that G is almost simple with socle X a finite simple group of Lie type, and several special cases where the Lie rank of X is at most $2$ have recently been settled in a sequence of papers. Our main theorem resolves the problem for all almost simple groups of Lie type, thereby completing the proof of Kantor's conjecture for generalized quadrangles with a flag-transitive automorphism group acting primitively on points and lines.
ER  -

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