Kernels of minimal characters of solvable groups

TL;DR

This paper investigates the structure of finite solvable groups, showing that certain quotient groups related to irreducible characters of minimal degree are nilpotent-by-abelian, revealing new insights into their character kernels.

Contribution

It proves that for finite solvable groups, the quotient by the kernel of an odd-degree minimal non-linear irreducible character is nilpotent-by-abelian.

Findings

G/{Ker} χ is nilpotent-by-abelian under given conditions

Character kernels relate to the group's nilpotent-by-abelian structure

Results apply to irreducible characters of minimal degree

Abstract

Let $G$ be a finite solvable group. We prove that if $\chi\in{\rm Irr}(G)$ has odd degree and $\chi(1)$ is the minimal degree of the non-linear irreducible characters of $G$, then $G/{\rm Ker} \chi$ is nilpotent-by-abelian.