Power graphs of (non)orientable genus two

TL;DR

This paper classifies finite groups based on whether their power graphs have orientable or nonorientable genus two, contributing to the understanding of the topological properties of power graphs.

Contribution

It provides a complete classification of finite groups with power graphs of (non)orientable genus two, a novel topological characterization.

Findings

Identifies all finite groups with power graphs of genus two

Distinguishes between orientable and nonorientable cases

Advances the topological understanding of power graphs

Abstract

The power graph $\Gamma_G$ of a finite group $G$ is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. In this paper, we classify the finite groups whose power graphs have (non)orientable genus two.