A group-theoretic condition equivalent to a condition on principal blocks

TL;DR

This paper establishes a group-theoretic criterion equivalent to a character-theoretic condition regarding the trivial character's uniqueness in principal p-blocks across specified primes in finite groups.

Contribution

It provides a new group-theoretic characterization that simplifies understanding when the trivial character is the only irreducible character in principal p-blocks for certain primes.

Findings

Equivalent group-theoretic and character-theoretic conditions identified

Simplifies analysis of principal p-blocks in finite groups

Connects group structure with block theory

Abstract

In this note, we give a group-theoretic condition which is equivalent to the fact that the trivial character is the only complex irreducible character of a finite group G which is contained in the principal p-block for each prime p in a specified set of prime divisors of the order of G. This character-theoretic condition has been studied previously by a number of authors.