Kernels of minimal characters of solvable groups

Abstract

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