Classification of Finite Groups with Toroidal or Projective-Planar Permutability Graphs

Abstract

Let G be a group. The permutability graph of subgroups of G, denoted by Γ(G), is a graph having all the proper subgroups of G as its vertices, and two subgroups are adjacent in Γ(G) if and only if they permute. In this article, 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_{1, 5}, P₅, P₆, C₆, or K_{3, 3} as a subgraph.

Key Words:

• Finite groups
• Genus
• Nonorientable genus
• Permutability graph
• Projective-planar graph
• Toroidal graph

2010 Mathematics Subject Classification:

• 05C25
• 05C10
• 20E34
• 57M07