A 3-Local Characterization of the Harada--Norton Sporadic Simple Group

TL;DR

This paper characterizes the Harada--Norton sporadic simple group and its automorphism group using 3-local subgroup structures, specifically focusing on the normalizer of the center of a Sylow 3-subgroup, contributing to the classification of sporadic groups.

Contribution

It provides a new 3-local characterization of the Harada--Norton group and its automorphism group based on their parabolic characteristic three structure.

Findings

Identified the groups from the normalizer of the center of a Sylow 3-subgroup.

Confirmed the groups are of parabolic characteristic three.

Established a new characterization method for these sporadic groups.

Abstract

We provide 3-local characterizations of the Harada--Norton sporadic simple group and its automorphism group. Both groups are examples of groups of parabolic characteristic three and we identify them from the structure of the normalizer of the centre of a Sylow 3-subgroup.