Finite groups whose subgroup graph contains a vertex of large degree

TL;DR

This paper investigates finite groups with a subgroup graph featuring a vertex of large degree, extending previous classifications to include solvable groups and revealing new structural insights.

Contribution

It extends the classification of finite groups with large degree vertices in their subgroup graphs to include solvable groups, building on prior work.

Findings

Classification of finite solvable groups with large degree vertices

Extension of Burness and Scott's results to solvable groups

Identification of structural properties related to subgroup graph degrees

Abstract

T.C. Burness and S.D. Scott \cite{3} classified finite groups $G$ such that the number of prime order subgroups of $G$ is greater than $|G|/2-1$. In this note, we study finite groups $G$ whose subgroup graph contains a vertex of degree greater than $|G|/2-1$. The classification given for finite solvable groups extends the work of Burness and Scott.