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

Christopher Herbig

TL;DR

This work analyzes generalized characters of finite groups, focusing on abelian groups whose values on nonidentity elements are sums of at most two roots of unity. The authors prove a near-complete structural description: such a generalized character on an abelian group typically has the form $\chi = a\rho_G + \delta_1\lambda_1 + \delta_2\lambda_2$, with $\delta_i\in\{-1,0,1\}$, except when $|G|$ is divisible by $2$, $3$, or $5$ where explicit outliers arise. The proof combines vanishing-sum results for roots of unity, Brauer’s characterization, and a detailed case analysis, yielding strong constraints on the prime divisors of $|G|$ and the constituents of $\chi$. As an application, the paper derives connectivity properties of prime graphs for groups admitting such characters, showing, for instance, that certain induced subgraphs become disconnected under natural prime-partitioning, while acknowledging that full disconnection of the entire prime graph remains delicate. Overall, the results advance the understanding of how constrained root-of-unity values in generalized characters influence group structure and prime-graph connectivity.

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.

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

TL;DR

This work analyzes generalized characters of finite groups, focusing on abelian groups whose values on nonidentity elements are sums of at most two roots of unity. The authors prove a near-complete structural description: such a generalized character on an abelian group typically has the form , with , except when is divisible by , , or where explicit outliers arise. The proof combines vanishing-sum results for roots of unity, Brauer’s characterization, and a detailed case analysis, yielding strong constraints on the prime divisors of and the constituents of . As an application, the paper derives connectivity properties of prime graphs for groups admitting such characters, showing, for instance, that certain induced subgraphs become disconnected under natural prime-partitioning, while acknowledging that full disconnection of the entire prime graph remains delicate. Overall, the results advance the understanding of how constrained root-of-unity values in generalized characters influence group structure and prime-graph connectivity.

Abstract

A character of a finite group having degree takes values which may be expressed as sums of 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.

Paper Structure

This paper contains 11 sections, 16 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of Main Result
  4. Case 1: $a_0 = 3$ and $a_2 = 0$, or vice versa.
  5. Case 2: $a_0 = 2$ and $a_2 = 1$, or vice versa.
  6. Case 3: $a_0 = 0$ and $a_2 = 4$, or vice versa.
  7. Case 4: $a_0 = 1$ and $a_2 = 3$, or vice versa.
  8. Case 5: $a_0 \geq 5$ or $a_2 \geq 5$.
  9. Prime Graphs
  10. Case 1: $\chi_H = a\rho_H \pm \lambda_1 \pm \lambda_2 \pm \lambda_3$ or $\chi_H = a\rho_H \pm \lambda_1 \pm \lambda_2 \pm \lambda_3 \pm \lambda_4$
  11. Case 2: $\chi_H = a\rho_H \pm \lambda_1 \pm \lambda_2 \mp \lambda_3$ or $\chi_H= a\rho_H \pm \lambda_1 \pm \lambda_2 \pm \lambda_3 \mp \lambda_4$.

Key Result

Theorem 1

Assume that the Sylow 2-subgroups of a finite group $G$ have no cyclic subgroups of index 2. Let $\chi$ be a generalized character taking root of unity values on $G^{\#}$. There exists a subset $\pi$ of $\pi(G)$ such that the restriction of $\chi$ to any $\pi$-subgroup [resp. $\pi'$-subgroup] $H$ ta

Theorems & Definitions (24)

  • Theorem 1: Robinson, 2011
  • Theorem 2: Robinson, 1999
  • Corollary 3
  • Corollary 4
  • proof
  • Theorem 5: Main Theorem
  • Proposition 6
  • proof
  • Lemma 8
  • Lemma 9
  • ...and 14 more