Finite Groups Whose Character Graphs Associated with Codegrees Have No Triangles

TL;DR

This paper characterizes finite groups with prime power irreducible character codegrees using new character graphs, classifying those whose codegree graphs lack triangles, extending previous work on degree-based character graphs.

Contribution

It introduces a novel type of character graph based on codegrees and classifies finite groups with triangle-free codegree graphs, advancing understanding of their structure.

Findings

Finite groups with prime power codegrees are characterized.

Classification of groups with triangle-free codegree graphs.

Connections established between codegree graphs and character coedgree graphs.

Abstract

Motivated by the Problem $164$ proposed by Y. Berkovich and E. Zhmud' in their book "Characters of finite groups, Part $1$", we give a characterization of finite groups whose irreducible character codegrees are prime powers. This is based on a new kind of character graphs of finite groups associated with codegrees. Such graphs have close and obvious connections with character coedgree graphs. For example, they have the same number of connected components. By analogy with the work of finite groups whose character graphs (associated with degrees) have no triangles, we conduct a result of classifying finite groups whose character graphs associated with codegrees have no triangles in the latter part of this paper.