Simple groups with Brauer trees of principal blocks in the shape of a star

TL;DR

This paper classifies finite simple groups with cyclic Sylow p-subgroups whose principal blocks have star-shaped Brauer trees, and provides necessary conditions for such structures in arbitrary groups.

Contribution

It identifies all simple groups with star-shaped Brauer trees in their principal blocks and establishes a necessary condition for any finite group with cyclic Sylow p-subgroups.

Findings

List of simple groups with star-shaped Brauer trees

Necessary condition for arbitrary groups with cyclic Sylow p-subgroups

Characterization of principal blocks with star-shaped Brauer trees

Abstract

We have found a list of finite simple groups with cyclic Sylow $p$-subgroup whose principal $p$-blocks have Brauer trees in the shape of a star, that is a tree of diameter at most $2$. Moreover, for an arbitrary finite group $G$ with cyclic Sylow $p$-subgroup, we have obtained a necessary condition when the Brauer tree of the principal $p$-block of $G$ is a star.