The commuting graph and a graph associated with centralizers

TL;DR

This work investigates how the commuting graph ${\mathfrak C}(G)$ of a finite group, especially a $p$-group, relates to the index $|G:Z(G)|$ and introduces a centralizer-graph ${\Gamma}_{\mathcal Z}(G)$ to study component structure and diameters. It establishes a web of correspondences among ${\mathfrak C}(G)$, its transversal variant ${\mathfrak C}^*(G)$, and the centralizer graph ${\Gamma}_{\mathcal Z}(G)$, showing that isoclinic groups yield isomorphic centralizer and transversal graphs, and that under certain order constraints one can bound connectivity and diameter. The paper also constructs families of groups (notably $G(p,n,S)$ and $\mathfrak G_p(n_1,...,n_k)$) with prescribed numbers of noncomplete components and diameters, thereby demonstrating how centralizer and commuting-graph structures can be engineered. Overall, it provides both general diameter/connectivity results and explicit combinatorial constructions linking group-theoretic centralizers to graph-theoretic features, enriching the toolkit for understanding noncommuting behavior in finite groups and enabling targeted SEO-friendly descriptors for graph-structured group data.

Abstract

Let $G$ be a $p$-group. We begin to consider the relationship between the structure of the commuting graph and $|G:Z(G)|$. We also build a family of groups whose commuting graphs have more than one connected component whose diameter is at least $2$. For this, we introduce another graph related to the commuting graph that is associated with centralizers.

Figures (1)

• Figure 1: Component $C_1$ of the centralizer graph $\Gamma_{\mathcal{Z}} (G)$ for $G={\mathfrak G}_3(3,n_2,\dots,n_k)$.

Theorems & Definitions (89)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Corollary 2.4
• proof