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Excluded power graphs of groups

Brian Curtin

TL;DR

This work investigates directed and undirected $\mathcal{X}$-excluded power graphs on groups, focusing on how excluding certain prime divisors reshapes the graph structure. It develops a quotient-graph framework and proves a key decomposition: for coprime $|H|$ and $|K|$, the $\pi$-excluded quotient power graph of a direct product $H\times K$ is a disjoint union of copies of the quotient power graph of $K$, indexed by cyclic subgroups of $H$. The authors then classify, for a fixed prime $p$, when the $\{p\}$-excluded graphs are disjoint unions of (directed) cliques, showing a precise semidirect-product structure under which this occurs. They further connect these results to normal Hall $\pi$-subgroups and provide explicit descriptions for nilpotent and dihedral-type groups, offering a structural lens into how cyclic subgroups govern excluded-power edges. The findings yield tractable decompositions that illuminate the interplay between group-theoretic decomposition (e.g., coprime factors, Hall subgroups, and semidirect products) and the topological features of the associated graphs.

Abstract

Let $G$ be a group, and let $\mathcal{X}$ be a set of integers greater than one. We consider the directed $\mathcal{X}$-excluded power graph of $G$, which takes $G$ as its vertex set and has an edge from $g$ to each power of $g$ other than itself provided the power is not divisible by any element of $\mathcal{X}$. We show that when $G=H\times K$ for $H$ and $K$ with coprime orders, excluding the prime factors of $|H|$ yields a power graph with a quotient consisting of multiple copies of a quotient of the power graph (no exclusions) of $K$. For any prime $p$, we describe the groups whose directed and undirected $\{p\}$-excluded power graphs consist of disjoint (directed) cliques.

Excluded power graphs of groups

TL;DR

This work investigates directed and undirected -excluded power graphs on groups, focusing on how excluding certain prime divisors reshapes the graph structure. It develops a quotient-graph framework and proves a key decomposition: for coprime and , the -excluded quotient power graph of a direct product is a disjoint union of copies of the quotient power graph of , indexed by cyclic subgroups of . The authors then classify, for a fixed prime , when the -excluded graphs are disjoint unions of (directed) cliques, showing a precise semidirect-product structure under which this occurs. They further connect these results to normal Hall -subgroups and provide explicit descriptions for nilpotent and dihedral-type groups, offering a structural lens into how cyclic subgroups govern excluded-power edges. The findings yield tractable decompositions that illuminate the interplay between group-theoretic decomposition (e.g., coprime factors, Hall subgroups, and semidirect products) and the topological features of the associated graphs.

Abstract

Let be a group, and let be a set of integers greater than one. We consider the directed -excluded power graph of , which takes as its vertex set and has an edge from to each power of other than itself provided the power is not divisible by any element of . We show that when for and with coprime orders, excluding the prime factors of yields a power graph with a quotient consisting of multiple copies of a quotient of the power graph (no exclusions) of . For any prime , we describe the groups whose directed and undirected -excluded power graphs consist of disjoint (directed) cliques.

Paper Structure

This paper contains 5 sections, 13 theorems, 7 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Excluded power graphs
  4. Semidirect product of groups of coprime order
  5. Directed cliques

Key Result

Lemma 2.2

c.f. MR3743245 Let $G$ be a group, and let $\mathcal{C}$ be the set of cyclic subgroups of $G$. In ${\mathcal{P}}_{}^{\begin{tikzpicture}{\draw[->](0,0)--(4pt,0);}\end{tikzpicture}}({G})$, $\mathrm{gen}({C})$ is a directed clique, and every element of $\mathrm{gen}({C})$ has the same in- and out-nei

Figures (5)

  • Figure 1: The excluded power graphs of $A_5$
  • Figure 2: Example of Corollary \ref{['cor:excludedindirectproduct']} with $\mathbb{Z}_{12}\cong\mathbb{Z}_3\times\mathbb{Z}_4$.
  • Figure 3: The power graph of a semidirect product $(\mathbb{Z}_3\times\mathbb{Z}_3)\rtimes_\psi \mathbb{Z}_2$
  • Figure 4: The power graph of a semidirect product $(\mathbb{Z}_3\times\mathbb{Z}_3)\rtimes_\psi \mathbb{Z}_2$
  • Figure 5: The $\{3\}$-excluded power graph of a semidirect product $\mathbb{Z}_7\rtimes_\phi\mathbb{Z}_3$

Theorems & Definitions (31)

  • Definition 1.1
  • Definition 1.2
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • ...and 21 more