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Ealy's conjecture in odd characteristic

Tao Feng, Koen Thas

TL;DR

This work resolves Ealy's conjecture for odd primes by proving that any finite thick generalized quadrangle where every point's central symmetries have order divisible by an odd prime $r$ must be one of the classical spheorgic types $\mathcal{W}(3,q)$, $\mathcal{H}(3,q^2)$, or $\mathcal{H}(4,q^2)$ with $q$ a power of $r$. It then classifies all thick GQs in which every point is a center of a nontrivial symmetry, yielding the same three geometries. The proofs combine synthetic incidence geometry with deep group-theoretic analysis: exceptional Lie-type socles are ruled out via order and subgroup constraints, while classical groups are handled through representation-theoretic and form-theoretic arguments; the generalized Ealy problem is settled without invoking CFSG in the geometric part. The results have applications to locally primitive GQs and provide a framework for addressing Kantor-type conjectures in this context.

Abstract

We solve Ealy's conjecture from 1977 by showing that for each odd prime $p$, a finite generalized quadrangle each point of which admits a central symmetry of order $p$, is either a classical symplectic quadrangle in dimension $3$, or a Hermitian quadrangle in dimension $3$ or $4$. As a byproduct, we vastly generalize the aforementioned result by determining the finite generalized quadrangles whose every point admits at least one nontrivial central symmetry.

Ealy's conjecture in odd characteristic

TL;DR

This work resolves Ealy's conjecture for odd primes by proving that any finite thick generalized quadrangle where every point's central symmetries have order divisible by an odd prime must be one of the classical spheorgic types , , or with a power of . It then classifies all thick GQs in which every point is a center of a nontrivial symmetry, yielding the same three geometries. The proofs combine synthetic incidence geometry with deep group-theoretic analysis: exceptional Lie-type socles are ruled out via order and subgroup constraints, while classical groups are handled through representation-theoretic and form-theoretic arguments; the generalized Ealy problem is settled without invoking CFSG in the geometric part. The results have applications to locally primitive GQs and provide a framework for addressing Kantor-type conjectures in this context.

Abstract

We solve Ealy's conjecture from 1977 by showing that for each odd prime , a finite generalized quadrangle each point of which admits a central symmetry of order , is either a classical symplectic quadrangle in dimension , or a Hermitian quadrangle in dimension or . As a byproduct, we vastly generalize the aforementioned result by determining the finite generalized quadrangles whose every point admits at least one nontrivial central symmetry.

Paper Structure

This paper contains 9 sections, 33 theorems, 21 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Description of the problem and notation
  3. The known classical GQs with a nontrivial symmetry
  4. Outline of the proof of the first main theorem
  5. Proof of the first main result: exceptional groups
  6. Proof of the first main result: classical groups
  7. The general Ealy problem
  8. Applications to locally primitive generalized quadrangles
  9. Appendix: Ealy's arguments

Key Result

Lemma 2.1

If $\mathcal{S}=(\mathcal{P},\mathcal{L})$ is a generalized quadrangle of order $(s,t)$, then

Theorems & Definitions (69)

  • Lemma 2.1
  • proof
  • Proposition 2.2: FGQ, 2.2.2 (dual version)
  • proof
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 59 more