Classification of finite groups with toroidal or projective-planar permutability graphs

TL;DR

This paper classifies finite groups based on the topological properties of their permutability graphs, specifically identifying those with toroidal or projective-planar graphs and those avoiding certain subgraphs.

Contribution

It provides a complete classification of finite groups with permutability graphs that are toroidal, projective-planar, or exclude specific subgraphs.

Findings

Finite groups with toroidal permutability graphs identified.

Finite groups with projective-planar permutability graphs classified.

Groups avoiding certain subgraphs like $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ characterized.

Abstract

Let $G$ be a group. The permutability graph of subgroups of $G$, denoted by $\Gamma(G)$, is a graph having all the proper subgroups of $G$ as its vertices, and two subgroups are adjacent in $\Gamma(G)$ if and only if they permute. In this paper, we classify the finite groups whose permutability graphs are toroidal or projective-planar. In addition, we classify the finite groups whose permutability graph does not contain one of $K_{3,3}$, $K_{1,5}$, $C_6$, $P_5$, or $P_6$ as a subgraph.