Finite groups whose commuting conjugacy class graphs have isolated vertices

Abstract

In this paper, we study a graph whose vertices are the non-trivial conjugacy classes of a finite group, and two conjugacy classes C₁ and C₂ are adjacent if and only if we can find $x\in C_1$ and $y\in C_2$ such that x, y commute. In particular, we investigate groups whose corresponding graphs contain isolated vertices. We classify solvable groups with this property and show that if G is non-solvable, then either $G/\mathbf{F}(G)$ is almost simple or G is sharply 2-transitive. We determine almost simple groups with sporadic socles and some families of simple groups with this property.

2020 Mathematics Subject Classification:

• Primary: 20D05

Keywords:

• Almost simple group
• fitting subgroup
• Frobenius group
• graph

Acknowledgments

The author is very much indebted to Professor Chris Parker for his helpful comments and suggestions which improved the results of this paper, especially by suggesting Proposition 3.9.