Excluded power graphs of groups

TL;DR

This paper introduces and studies a new class of power graphs called $\\mathcal{X}$-excluded power graphs, analyzing their structure, especially for product groups and semidirect products, and characterizing groups with disjoint clique structures.

Contribution

It defines the $\\mathcal{X}$-excluded power graph, explores their properties for product and semidirect product groups, and characterizes groups with disjoint clique $\\mathcal{X}$-excluded power graphs.

Findings

Quotient structure of the $\\mathcal{X}$-excluded power graph for product groups.

Partial results on semidirect product groups.

Characterization of groups with disjoint clique $\\mathcal{X}$-excluded power graphs.

Abstract

Let $\mathcal{X}$ be a set of integers greater than one. The $\mathcal{X}$-excluded power graph of a group $G$ has vertex set $G$ and an edge from $g$ to each power of $g$ other than itself provided that the power is not divisible by any element of $\mathcal{X}$. When $G=H\times K$ for groups $H$ 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 $K$. Partial results for the semidirect product under the same conditions are given. We describe groups whose $\mathcal{X}$-excluded power graphs consist of disjoint directed cliques.