A supercharacter analog of vanishing-off subgroups and generalized Camina pair

TL;DR

This work extends vanishing-off subgroups and Camina-type concepts to the supercharacter theory setting by introducing $S$-GCP, $S$-Camina pairs, and the subgroups $V(S)$ and $U(S|N)$. It establishes several equivalent characterizations of $S$-GCPs, analyzes their interaction with $[G,S]$, $Z(S)$, and the upper central series, and connects vanishing properties to $\ast$- and $\Delta$-product constructions, yielding criteria for $S$-nilpotence and explicit degree structures in $VZ(S)$-groups. The results build a structural bridge between classical Camina-type ideas and supercharacter theory, highlighting how product decompositions control vanishing behavior and central-series containment. Together, these contributions provide a framework for translating key classical group-theoretic insights into the language of supercharacters and offer tools for analyzing $S$-abelian versus non-$S$-abelian groups through vanishing phenomena.

Abstract

Vanishing-off subgroups, generalized Camina pair and other related subgroups have played a significant role in the study of group structure. The primary goal of this paper is to study their analogs in the setting of supercharacter theory. We establish several properties of these subgroups which includes connections with supercharacter theory products.

Theorems & Definitions (63)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Theorem 2.1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Theorem 2.2