Classification of non-solvable groups whose power graph is a cograph

TL;DR

This paper classifies finite groups with power graphs that are cographs, focusing on non-solvable groups related to PSL 2(q) or Sz(q), and provides a detailed structure for solvable cases, linking to open problems in automorphism groups.

Contribution

It completes the classification of non-solvable power-cograph groups by addressing number theoretic obstacles and describes solvable power-cograph groups, especially with connected Gruenberg-Kegel graphs.

Findings

Classification of non-solvable power-cograph groups is achieved.

Complete description of solvable power-cograph groups with connected Gruenberg-Kegel graph.

Reduction of disconnected cases to open problems in p-group automorphisms.

Abstract

Cameron, Manna and Mehatari investigated the question of which finite groups admit a power graph that is a cograph, also called power-cograph groups (Journal of Algebra 591 (2022)). They give a classification for nilpotent groups and partial results for general groups. However, the authors point out number theoretic obstacles towards a classification. These arise when the groups are assumed to be isomorphic to PSL 2 (q) or Sz(q) and are likely to be hard. In this paper, we prove that these number theoretic problems are in fact the only obstacles to the classification of non-solvable power-cograph groups. Specifically, for the non-solvable case, we give a classification of power-cograph groups in terms of such groups isomorphic to PSL 2 (q) or Sz(q). For the solvable case, we are able to precisely describe the structure of solvable power-cograph groups. We obtain a complete classification of solvable power-cograph groups whose Gruenberg-Kegel graph is connected. Moreover, we reduce the case where the Gruenberg-Kegel graph is disconnected to the classification of p-groups admitting fixed-point-free automorphisms of prime power order, which is in general an open problem.