On Generalized Characters Whose Values on Nonidentity Elements are Sums of at Most Two Roots of Unity

TL;DR

This paper characterizes irreducible constituents of generalized characters on abelian groups with values as sums of two or fewer roots of unity, and explores implications for prime graph connectivity in such groups.

Contribution

It provides a new description of generalized characters with restricted root of unity sums and applies this to analyze prime graph connectivity in finite groups.

Findings

Irreducible constituents are characterized for characters with values as sums of two roots of unity.

Main result links character values to group structure and prime graph properties.

Applications include insights into the connectivity of prime graphs for certain finite groups.

Abstract

A character of a finite group having degree $n$ takes values which may be expressed as sums of $n$ or fewer roots of unity. In this note, we prove a result which describes the irreducible constituents of generalized characters on abelian groups whose values on nonidentity elements are expressible as sums of two or fewer roots of unity. In Section 4, we apply our main result to obtain information about the connectivity of prime graphs for groups admitting such characters.