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PreprintCC BY 4.0arXiv

The Siemons--Wagner property and Bradley's conjecture

  • Zhilin Zhang
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Abstract

A transitive permutation group G≤Sym(Pcal) is said to have the Siemons--Wagner property ⋆3 if some three-point subset has a G-orbit longer than the orbit of every four-point subset containing it. In this paper we develop a structural theory of three-point orbits of finite $2$-transitive groups (orbit-length hierarchies, pointwise and setwise stabilizers, and their geometric realisations, including the unital and the ovoid), and we use it to completely settle Bradley's conjecture: a finite primitive group of degree at least $8$ satisfying ⋆3 is isomorphic to PSL(2,7) or PGL(2,7).

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026siemonswagner,
  title = {{The Siemons--Wagner property and Bradley's conjecture}},
  author = {Zhilin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24502},
  eprint = {2610.24502},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24502},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Siemons--Wagner property and Bradley's conjecture
AU  - Zhilin Zhang
PY  - 2026
DA  - 2026/10/17
PB  - arXiv
DO  - 10.48550/arxiv.2610.24502
UR  - https://arxiv.org/abs/2610.24502
AB  - A transitive permutation group G≤Sym(Pcal) is said to have the Siemons--Wagner property ⋆_3 if some three-point subset has a G-orbit longer than the orbit of every four-point subset containing it. In this paper we develop a structural theory of three-point orbits of finite $2$-transitive groups (orbit-length hierarchies, pointwise and setwise stabilizers, and their geometric realisations, including the unital and the ovoid), and we use it to completely settle Bradley's conjecture: a finite primitive group of degree at least $8$ satisfying ⋆_3 is isomorphic to PSL(2,7) or PGL(2,7).
ER  -