The continuity of $p$-rationality and a lower bound for $p'$-degree characters of finite groups

TL;DR

This paper establishes a new bound on the number of certain irreducible characters of finite groups based on the structure of Sylow p-subgroups, confirming a continuity property for p=2 linked to longstanding conjectures.

Contribution

It provides a strong bound for p'-degree irreducible characters using the commutator factor group and proves the continuity of p-rationality levels for p=2, connecting to the McKay-Navarro conjecture.

Findings

Established a bound for p'-degree characters in terms of Sylow p-subgroup structure.

Proved the continuity of p-rationality levels for p=2.

Linked the results to the McKay-Navarro conjecture.

Abstract

Let $p$ be a prime and $G$ a finite group. We propose a strong bound for the number of $p'$-degree irreducible characters of $G$ in terms of the commutator factor group of a Sylow $p$-subgroup of $G$. The bound arises from a recent conjecture of Navarro and Tiep [NT21] on fields of character values and a phenomenon called the continuity of $p$-rationality level of $p'$-degree characters. This continuity property in turn is predicted by the celebrated McKay-Navarro conjecture [Nav04]. We achieve both the bound and the continuity property for $p=2$.