On finite generalized quadrangles of even order

TL;DR

This paper proves two conjectures about finite generalized quadrangles of even order: that skew translation ones are translation and that they lack point regular automorphism groups, advancing understanding of their symmetry properties.

Contribution

It confirms Payne's conjecture on skew translation generalized quadrangles and Ghinelli's conjecture on automorphism groups for even order cases.

Findings

Skew translation generalized quadrangles of even order are translation generalized quadrangles.

Generalized quadrangles of even order do not admit point regular automorphism groups.

Confirmed two longstanding conjectures in finite geometry.

Abstract

In this paper, we establish the following two results: (1) a skew translation generalized quadrangle of even order is a translation generalized quadrangle, (2) a generalized quadrangle of even order does not admit a point regular automorphism group. The first result confirms a conjecture of Payne (1975) based on earlier work of Ott (2021), and the second result confirms a conjecture of Ghinelli (1992).