On almost $p$-rational characters in principal blocks

TL;DR

This paper establishes a lower bound on the number of almost p-rational characters with degree coprime to p in the principal p-block of finite groups, and characterizes the groups where this bound is achieved.

Contribution

It introduces a new lower bound for almost p-rational characters in principal p-blocks and describes the p-local structure of groups attaining this bound.

Findings

Lower bound for the number of almost p-rational characters

Characterization of groups where the bound is sharp

Insights into the p-local structure of these groups

Abstract

Let p be a prime. In this paper we provide a lower bound for the number of almost p-rational characters of degree coprime to p in the principal p-block of a finite group of order divisible by p. We further describe the p-local structure of the groups for which the above-mentioned bound is sharp.