Groups whose character degree graph has diameter three

TL;DR

This paper characterizes finite solvable groups whose prime graph of character degrees has the maximum diameter of three, confirming related conjectures and deepening understanding of the structure of such groups.

Contribution

It provides a complete description of solvable groups with prime graphs of diameter three, advancing the classification and confirming existing conjectures.

Findings

Characterization of solvable groups with diameter three prime graphs

Confirmation of conjectures by M.L. Lewis

Enhanced understanding of the structure of character degree graphs

Abstract

Let \(G\) be a finite group, and let \(\Delta(G)\) denote the \emph{prime graph} built on the set of degrees of the irreducible complex characters of \(G\). It is well known that, whenever \(\Delta(G)\) is connected, the diameter of \(\Delta(G)\) is at most \(3\). In the present paper, we provide a description of the finite solvable groups for which the diameter of this graph attains the upper bound. This also enables us to confirm a couple of conjectures proposed by M.L. Lewis.