Ealy's conjecture in odd characteristic

TL;DR

This paper proves Ealy's conjecture for odd primes, classifying finite generalized quadrangles with points admitting central symmetries, and extends the classification to those with at least one nontrivial central symmetry per point.

Contribution

It confirms Ealy's conjecture for all odd primes and generalizes the classification of generalized quadrangles with central symmetries at each point.

Findings

Finite generalized quadrangles with central symmetries are either classical symplectic or Hermitian quadrangles.

The classification extends to quadrangles where each point admits at least one nontrivial central symmetry.

The result confirms the structure of such quadrangles in odd characteristic.

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.