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Unitals without O'Nan configurations are classical if they admit all translations

Markus Johannes Stroppel

Abstract

We prove the statement in the title: if a (finite) unital admits all translations and contains no O'Nan configurations then the unital is classical, i.e., isomorphic to the Hermitian unital of the same order.

Unitals without O'Nan configurations are classical if they admit all translations

Abstract

We prove the statement in the title: if a (finite) unital admits all translations and contains no O'Nan configurations then the unital is classical, i.e., isomorphic to the Hermitian unital of the same order.

Paper Structure

This paper contains 2 sections, 3 theorems, 2 equations, 2 figures.

Table of Contents

  1. Unitals, translations
  2. Wilbrink's conditions

Key Result

Lemma 1.2

For each point $c$ of $\mathcal{U}$, the set $T_{[c]}$ of all translations with center $c$ forms a subgroup of $\operatorname{Aut}_{}(\mathcal{U})$ that acts semi-regularly on $P\smallsetminus\{c\}$.

Figures (2)

  • Figure 1: An O'Nan configuration.
  • Figure 2: Wilbrink's condition (III)

Theorems & Definitions (4)

  • Lemma 1.2: MR3090721
  • Theorem 2.2: MR690826
  • Theorem 2.3
  • proof