Vanishing-set inclusion and character degrees in finite $p$-groups

Abstract

Let $G$ be a finite $p$-group and let $χ,ψ\in\operatorname{Irr}(G)$. We prove that $$\operatorname{Van}(χ)\subseteq\operatorname{Van}(ψ) \quad\Longrightarrow\quad χ(1)\midψ(1),$$ where $\operatorname{Van}(θ)=\{g\in G:θ(g)=0\}$. Consequently, two irreducible characters of a finite $p$-group with the same vanishing set have the same degree. This answers a question attributed to J. Sangroniz and recorded by Hung, Moretó, and Morotti, extending the previously known case in which one character has degree $p$. The same divisibility conclusion holds for irreducible characters of finite nilpotent groups.