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Characterizing finite groups whose enhanced power graphs have universal vertices

David G. Costanzo, Mark L. Lewis, Stefano Schmidt, Eyob Tsegaye, Gabe Udell

Abstract

Let $G$ be a finite group and construct a graph $Δ(G)$ by taking $G\setminus\{1\}$ as the vertex set of $Δ(G)$ and by drawing an edge between two vertices $x$ and $y$ if $\langle x,y\rangle$ is cyclic. Let $K(G)$ be the set consisting of the universal vertices of $Δ(G)$ along the identity element. For a solvable group $G$, we present a necessary and sufficient conditon for $K(G)$ to be nontrivial. We also develop a connection between $Δ(G)$ and $K(G)$ when $|G|$ is divisible by two distinct primes and the diameter of $Δ(G)$ is $2$.

Characterizing finite groups whose enhanced power graphs have universal vertices

Abstract

Let be a finite group and construct a graph by taking as the vertex set of and by drawing an edge between two vertices and if is cyclic. Let be the set consisting of the universal vertices of along the identity element. For a solvable group , we present a necessary and sufficient conditon for to be nontrivial. We also develop a connection between and when is divisible by two distinct primes and the diameter of is .
Paper Structure (3 sections, 7 theorems, 1 equation)

This paper contains 3 sections, 7 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Solvable groups with $K(G)$ nontrivial
  3. $\{p,q\}$-groups

Key Result

Theorem A

Let $G$ be a solvable group, let $p$ be a prime, and let $P$ be a Sylow $p$-subgroup of $G$. Then $p$ is a prime divisor of $|K(G)|$ if and only if one of following situations occurs: If conclusion (2) occurs, then $|K(G)|_{2}=2$.

Theorems & Definitions (14)

  • Theorem A
  • Theorem B
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['K(G) main']}
  • Lemma 3.1
  • ...and 4 more