Square character degree graphs yield direct products

TL;DR

This paper proves that if a solvable group's character degree graph forms a square, then the group is necessarily a direct product of smaller groups.

Contribution

It establishes a novel link between the geometric shape of the character degree graph and the algebraic structure of the group.

Findings

Square character degree graphs imply the group is a direct product.

Provides a characterization of solvable groups based on their character degree graphs.

Enhances understanding of the relationship between graph structure and group decomposition.

Abstract

If $G$ is a solvable group, we take $\Delta (G)$ to be the character degree graph for $G$ with primes as vertices. We prove that if $\Delta (G)$ is a square, then $G$ must be a direct product.