Classifying Camina groups: a theorem of Dark and Scoppola

TL;DR

This paper offers a new proof for the classification of Camina groups, strengthening existing results and providing insights into their structure through an improved theorem related to Camina pairs.

Contribution

It presents a novel proof of the classification of Camina groups using a strengthened theorem of Isaacs, which is of independent mathematical interest.

Findings

New proof of Camina groups classification

Strengthened theorem of Isaacs on Camina pairs

Enhanced understanding of Camina group structure

Abstract

Recall that a group $G$ is a Camina group if every nonlinear irreducible character of $G$ vanishes on $G \setminus G'$. Dark and Scoppola classified the Camina groups that can occur. We present a different proof of this classification using Theorem 2, which strengthens a result of Isaacs on Camina pairs. Theorem 2 is of independent interest.