On the regularity of a graph related to conjugacy class sizes of a normal subgroup

TL;DR

This paper investigates the structure of a specific graph related to conjugacy class sizes in a finite group with a normal subgroup, proving that certain regularity conditions imply a particular direct product decomposition of the subgroup.

Contribution

It characterizes the structure of the normal subgroup when the associated conjugacy class size graph is connected, incomplete, and regular, revealing a specific direct product form.

Findings

If the graph is connected, incomplete, and regular, then the subgroup decomposes as a p-group times an abelian subgroup.

The subgroup's center is not equal to its intersection with the group's center under these conditions.

The graph's regularity imposes strong structural constraints on the subgroup.

Abstract

Given a finite group $G$ with a normal subgroup $N$, the simple graph $\Gamma_\textit{G}( \textit{N} )$ is a graph whose vertices are of the form $|x^G|$, where $x\in{N\setminus{Z(G)}}$, and $x^G$ is the $G$-conjugacy class of $N$ containing the element $x$. Two vertices $|x^G|$ and $|y^G|$ are adjacent if they are not co-prime. In this article we prove that, if $\Gamma_G(N)$ is a connected incomplete regular graph, then $N= P \times{A}$ where $P$ is a $p$-group, for some prime $p$ and $A\leq{Z(G)}$, and ${\bf Z}(N)\not = N\cap {\bf Z}(G)$.