This dissertation explores two constructions of loop extensions: Schreier extensions of Steiner loops and a new extension formula for right Bol loops arising from Bol reflections.Steiner loops are a key tool in studying Steiner triple systems. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provides a powerful method for constructing Steiner triple systems containing Veblen points. We determine the number of the Steiner triple systems of sizes 19, 27 and 31 with Veblen points, presenting some examples.Furthermore, we study a new extension formula for right Bol loops. We prove the necessary and sufficient conditions for the extension to be right Bol as well. We describe the most important invariants: right multiplication group, nuclei, center. We show that the core is an involutory quandle which is the disjoint union of two isomorphic involutory quandles. Lastly, we derive further results on the structure group of the core of the extension.

(2024). Schreier extensions of Steiner loops and extensions of Bol loops arising from Bol reflections.

### Schreier extensions of Steiner loops and extensions of Bol loops arising from Bol reflections

#### Abstract

This dissertation explores two constructions of loop extensions: Schreier extensions of Steiner loops and a new extension formula for right Bol loops arising from Bol reflections.Steiner loops are a key tool in studying Steiner triple systems. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provides a powerful method for constructing Steiner triple systems containing Veblen points. We determine the number of the Steiner triple systems of sizes 19, 27 and 31 with Veblen points, presenting some examples.Furthermore, we study a new extension formula for right Bol loops. We prove the necessary and sufficient conditions for the extension to be right Bol as well. We describe the most important invariants: right multiplication group, nuclei, center. We show that the core is an involutory quandle which is the disjoint union of two isomorphic involutory quandles. Lastly, we derive further results on the structure group of the core of the extension.
##### Scheda breve Scheda completa Scheda completa (DC)
feb-2024
Steiner triple systems; loops; Steiner loops; Veblen points; extensions; Bol loops; Bol reflections; quandles
(2024). Schreier extensions of Steiner loops and extensions of Bol loops arising from Bol reflections.
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Descrizione: Schreier extensions of Steiner loops and extensions of Bol loops arising from Bol reflections
Tipologia: Tesi di dottorato
Dimensione 1.4 MB
Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10447/623601`