On groups with chordal power graph, including a classification in the case of finite simple groups

TL;DR

This paper investigates the structure of groups with chordal power graphs, providing classifications for finite simple groups and other classes, and establishing bounds on induced path lengths to aid further classification efforts.

Contribution

It classifies finite simple groups with chordal power graphs and introduces conditions for cycle existence, expanding understanding beyond previously classified nilpotent groups.

Findings

Finite simple groups with chordal power graphs are classified.

Conditions for the existence of long cycles in power graphs are established.

A bound on the length of induced paths in chordal power graphs is provided.

Abstract

We prove various properties on the structure of groups whose power graph is chordal. Nilpotent groups with this property have been classified by Manna, Cameron and Mehatari [The Electronic Journal of Combinatorics, 2021]. Here we classify the finite simple groups with chordal power graph, relative to typical number theoretic oracles. We do so by devising several sufficient conditions for the existence and non-existence of long cycles in power graphs of finite groups. We examine other natural group classes, including special linear, symmetric, generalized dihedral and quaternion groups, and we characterize direct products with chordal power graph. The classification problem is thereby reduced to directly indecomposable groups and we further obtain a list of possible socles. Lastly, we give a general bound on the length of an induced path in chordal power graphs, providing another potential road to advance the classification beyond simple groups.