Blocks with only one irreducible Brauer character orbit

TL;DR

This paper proves that certain blocks with abelian defect groups are inertial if they cover a specific type of block with a single Brauer character orbit in a normal subgroup.

Contribution

It establishes a new criterion linking the structure of blocks with abelian defect groups to the orbit structure of Brauer characters in normal subgroups.

Findings

Blocks with abelian defect groups are inertial under the given conditions.

The paper connects the orbit structure of Brauer characters to block inertiality.

Provides a new perspective on the structure of blocks in modular representation theory.

Abstract

In this paper, we prove that a \(p\)-block with abelian defect group is inertial if it covers a \(p\)-block of a normal subgroup of \(p\)-power index having only one irreducible Brauer character orbit.