Characterizing finite groups whose enhanced power graphs have universal vertices

Fuente: arXiv
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Autores principales: Costanzo, David G., Lewis, Mark L., Schmidt, Stefano, Tsegaye, Eyob, Udell, Gabe
Formato: Preprint
Publicado: 2024
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author Costanzo, David G.
Lewis, Mark L.
Schmidt, Stefano
Tsegaye, Eyob
Udell, Gabe
author_facet Costanzo, David G.
Lewis, Mark L.
Schmidt, Stefano
Tsegaye, Eyob
Udell, Gabe
contents 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 $2$.
format Preprint
id arxiv_https___arxiv_org_abs_2402_06157
institution arXiv
publishDate 2024
record_format arxiv
spellingShingle Characterizing finite groups whose enhanced power graphs have universal vertices
Costanzo, David G.
Lewis, Mark L.
Schmidt, Stefano
Tsegaye, Eyob
Udell, Gabe
Group Theory
Primary 20D25, Secondary 05C25
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 $2$.
title Characterizing finite groups whose enhanced power graphs have universal vertices
topic Group Theory
Primary 20D25, Secondary 05C25
url https://arxiv.org/abs/2402.06157