Towards the standard Majorana representations of 3-transposition groups

TL;DR

This paper investigates 3-transposition groups, identifies sizes of their maximal symmetric subgroups, and constructs faithful representations in various spaces to propose candidates for standard Majorana representations.

Contribution

It introduces a method to find candidate groups for standard Majorana representations by analyzing subgroup sizes and constructing faithful representations in orthogonal, symplectic, and unitary spaces.

Findings

Identified sizes of maximal symmetric subgroups in Fischer groups

Constructed faithful representations in multiple geometric spaces

Proposed candidate groups for standard Majorana representations

Abstract

In this paper, we discuss 3-transposition groups. In particular, we find sizes of maximal symmetric subgroups of the groups, which are in Fischer list. In addition, we build faithful representations of symmetric groups in orthogonal, symplectic, and unitary spaces, which fully verify the main result. Thus, we find candidate groups, for which one may be able to build standard Majorana representations.