Character degrees in $\pi$-separable groups

TL;DR

This paper extends classical theorems in character theory to $ ext{pi}$-separable groups by involving specific sets of irreducible characters, providing new insights into their structure and properties.

Contribution

It introduces variants of the Ito-Michler and Thompson theorems for $ ext{pi}$-separable groups using the sets of irreducible characters $ ext{B}_ ext{pi}(G)$ and $ ext{B}_{ ext{pi}'}(G)$, advancing the understanding of character degrees.

Findings

Variants of Ito-Michler theorem for $ ext{pi}$-separable groups.

Variants of Thompson's theorem involving $ ext{B}_ ext{pi}(G)$ and $ ext{B}_{ ext{pi}'}(G)$.

New character-theoretic results for $ ext{pi}$-separable groups.

Abstract

If a group $G$ is $\pi$-separable, where $\pi$ is a set of primes, the set of irreducible characters $\operatorname{B}_{\pi}(G) \cup \operatorname{B}_{\pi'}(G)$ can be defined. In this paper, we prove that there are variants of some classical theorems in character theory, namely the Theorem of Ito-Michler and Thompson theorem on character degrees, which involve irreducible characters in the set $\operatorname{B}_{\pi}(G) \cup \operatorname{B}_{\pi'}(G)$.