On the Genus of the Nilpotent Graphs of Finite Groups

Abstract

The nilpotent graph of a group G is a simple graph whose vertex set is G∖nil(G), where nil(G) = {y ∈ G | ⟨ x, y ⟩ is nilpotent ∀ x ∈ G}, and two distinct vertices x and y are adjacent if ⟨ x, y ⟩ is nilpotent. In this article, we show that the collection of finite non-nilpotent groups whose nilpotent graphs have the same genus is finite, derive explicit formulas for the genus of the nilpotent graphs of some well-known classes of finite non-nilpotent groups, and determine all finite non-nilpotent groups whose nilpotent graphs are planar or toroidal.

Key Words:

• Finite group
• Nilpotent graph

2010 Mathematics Subject Classification:

• Primary: 20D60
• Secondary: 05C25

ACKNOWLEDGMENTS

Both authors are extremely grateful to the referee for the invaluable help and suggestions. The second author is also grateful to Professor Derek Holt of University of Warwick for providing a direct proof of Lemma 3.1(b) through math.stackexchange.

Notes

Communicated by D. Macpherson.