Minimal 3-generated Majorana algebras

TL;DR

This paper introduces minimal 3-generated Majorana algebras, classifies related 6-transposition groups, and uses computational methods to describe the algebras derived from these groups, advancing understanding of Majorana theory.

Contribution

It defines minimal 3-generated Majorana algebras, classifies associated 6-transposition groups, and employs computational tools to analyze the resulting algebras, providing new insights into their structure.

Findings

Complete classification of finite minimal 3-generated 6-transposition groups.

Almost complete description of minimal 3-generated Majorana algebras from these groups.

Development of computational methods for analyzing Majorana algebras.

Abstract

Majorana theory was introduced by A. A. Ivanov as the axiomatization of certain properties of the 2A-axes of the Griess algebra. Since its inception, Majorana theory has proved to be a remarkable tool with which to study objects related to the Griess algebra and the Monster simple group. We introduce the definition of a minimal 3-generated Majorana algebra and begin the first steps towards classifying such algebras. In particular, we give a complete classification of finite minimal 3-generated 6-transposition groups. We then use an algorithm developed in GAP by M. Pfeiffer and M. Whybrow, together with some additional computational tools, to give an almost complete description of all minimal 3-generated Majorana algebras arising from this list of groups.