Defect zero characters predicted by local structure

TL;DR

This paper investigates conditions under which finite groups have defect zero blocks, focusing on prime divisibility properties and extending known results for specific primes.

Contribution

It generalizes the existence of defect zero blocks in finite groups based on prime divisibility conditions, extending classical results.

Findings

If a prime q divides |G| but not any p-local subgroup, then G has a p-block of defect zero under certain conditions.

The result extends the classical case q=2 by Brauer and Fowler.

Provides new criteria for the existence of defect zero blocks in finite groups.

Abstract

Let $G$ be a finite group and let $p$ be a prime. Assume that there exists a prime $q$ dividing $|G|$ which does not divide the order of any $p$-local subgroup of $G$. If $G$ is $p$-solvable or $q$ divides $p-1$, then $G$ has a $p$-block of defect zero. The case $q=2$ is a well-known result by Brauer and Fowler.