On finite groups whose coprime graph is a divisor graph

TL;DR

This paper characterizes when coprime graphs of finite groups are divisor graphs, providing classifications for various group types and establishing connections with generalized lexicographic products.

Contribution

It introduces a characterization of divisor graphs among generalized lexicographic products and applies this to classify finite groups with divisor coprime graphs.

Findings

Power, reduced power, and order graphs are divisor graphs.

Coprime graphs of certain groups are divisor graphs, including nilpotent, dihedral, and symmetric groups.

Classification of finite groups with at most four prime divisors whose coprime graphs are divisor graphs.

Abstract

In this paper, we first characterize which generalized lexicographic products are divisor graphs. As applications, we show that power graphs, reduced power graphs and order graphs are all divisor graphs, which also implies the main result in [Power graph of a finite group is always divisor graph, Asian-European Journal of Mathematics 16 (2023)]. We then show that, the coprime graph of a group is a generalized lexicographic product, and characterize which coprime graphs are divisor graphs. Finally, we classify the finite groups $G$ having at most four prime divisors, whose coprime graphs are divisor graphs, and we also classify the finite groups $G$ whose coprime graphs are divisor graphs, if $G$ is a nilpotent group, a dihedral group, a generalized quaternion group, a symmetric group, an alternating group, a direct product of two non-trivial groups, and a sporadic simple group.