On twisted conjugacy classes of type D in sporadic simple groups

TL;DR

This paper classifies twisted conjugacy classes of type D in sporadic simple groups and shows that certain finite-dimensional pointed Hopf algebras are group algebras, advancing the classification of these algebraic structures.

Contribution

It provides the first classification of twisted conjugacy classes of type D in sporadic simple groups and proves that specific Hopf algebras are group algebras.

Findings

Classified twisted conjugacy classes of type D in all sporadic simple groups.

Proved that over certain groups, all finite-dimensional pointed Hopf algebras are group algebras.

Enhanced understanding of conjugacy classes of type D in sporadic simple groups.

Abstract

We classify twisted conjugacy classes of type D associated to the sporadic simple groups. This is an important step in the program of the classification of finite-dimensional pointed Hopf algebras with non-abelian coradical. As a by-product we prove that every complex finite-dimensional pointed Hopf algebra over the group of automorphisms of M12, J2, Suz, He, HN, T is the group algebra. In the appendix we improve the study of conjugacy classes of type D in sporadic simple groups.