The connectivity of the normalising and permuting graph of a finite soluble group

TL;DR

This work analyzes two graphs defined on finite soluble groups: the normalising graph $\\Gamma(G)$ and the permuting graph $\\Psi(G)$. The authors classify when $\\Gamma(G)$ is disconnected (precisely for Frobenius groups $G=KC$ with a prime-division condition) and show that, when connected, $\\Gamma(G)$ has diameter at most $6$ (a bound shown to be tight); they prove $\\Psi(G)$ is connected exactly when $\\Gamma(G)$ is connected and inherits the same diameter bound, with a diameter-$6$ example. A key intermediate result is that the distance from any nontrivial element to a minimal normal subgroup is bounded unless the group is Frobenius, which underpins the diameter bounds. Finally, the paper provides a concrete soluble group with $\\Psi(G)$ (and hence $\\Gamma(G)$) achieving diameter $6$, establishing tightness of the main theorems and linking the graph structure to Frobenius group theory.

Abstract

We introduce the normalising graph of a group and study the connectivity of the normalising and permuting graphs of a group when the group is finite and soluble. In particular, we classify finite soluble groups with disconnected normalising graph. The main results shows that if a finite soluble group has connected normalising graph then this graph has diameter at most 6. Furthermore, this bound is tight. A corollary then presents the connectivity properties of the permuting graph.

Theorems & Definitions (34)

• Theorem 1
• Corollary 2
• proof
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Lemma 6