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arXiv · 2611.03541

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

Zhi-Lin Zhang
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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.

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Zhi-Lin Zhang. 2026-11-02. Vanishing-set inclusion and character degrees in finite $p$-groups. https://arxiv.org/abs/2611.03541

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