Solvable conjugacy class graph of groups

TL;DR

This paper introduces the solvable conjugacy class graph (SCC-graph) for groups, analyzing its properties and classifying finite groups with specific clique numbers, notably providing a complete list for clique number 2.

Contribution

It defines the SCC-graph for groups, studies its properties, and classifies finite groups with a given clique number, especially for clique number 2.

Findings

Finite groups with a given SCC-graph clique number are finite in number.

Explicit classification of groups with SCC-graph clique number 2.

Properties such as connectivity, girth, and clique number of SCC-graphs are analyzed.

Abstract

In this paper we introduce the graph $\Gamma_{sc}(G)$ associated with a group $G$, called the solvable conjugacy class graph (abbreviated as SCC-graph), whose vertices are the nontrivial conjugacy classes of $G$ and two distinct conjugacy classes $C, D$ are adjacent if there exist $x \in C$ and $y \in D$ such that $\langle x, y\rangle$ is solvable. We discuss the connectivity, girth, clique number, and several other properties of the SCC-graph. One of our results asserts that there are only finitely many finite groups whose SCC-graph has given clique number~$d$, and we find explicitly the list of such groups with $d=2$.