# Vanishing-off subgroups and supercharacter theory products

## TL;DR

This paper explores the properties of vanishing-off subgroups in supercharacter theory and introduces new characterizations for supercharacter theory products, enabling determination of product types from supercharacter values.

## Contribution

It provides novel characterizations of supercharacter theory products using vanishing-off subgroups and supercharacter values, advancing understanding of supercharacter theory structure.

## Key findings

- Characterization of supercharacter theory products via supercharacter values
- Identification of conditions for $ riangle$- and $igast$-products
- Enhanced methods to determine supercharacter theory structures

## Abstract

In this paper, we study the vanishing-off subgroups of supercharacters, and use these to determine several new characterizations of supercharacter theory products. In particular, we give a character theoretic characterization that allows us to conclude that one may determine if a supercharacter theory is a $\Delta$-product or $\ast$-product from the values of its corresponding supercharacters.