Finite groups whose subgroup graph contains a vertex of large degree

TL;DR

This paper extends the Burness–Scott classification by determining finite solvable groups whose subgroup graph $L(G)^*$ contains a vertex of degree greater than $|G|/2-1$. It develops two key tools: a general degree bound for a vertex in $L(G)^*$ and a bound on maximal subgroups, then uses a case-analysis driven by these bounds to constrain the structure of $G$. The main result identifies explicit families, including elementary abelian $2$-groups, generalized extraspecial $2$-groups, certain semidirect products $C_p^n \rtimes C_2$, and the dihedral group $D_{12}$, along with small-order exceptions and the groups of types (I)-(IX) from Theorem A. The approach reveals that a large-degree vertex in the subgroup graph must lie at the top or bottom of the subgroup lattice in solvable groups, contributing to a refined understanding of extremal subgroup-graph configurations.

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.

Theorems & Definitions (10)

• Theorem A
• Theorem 1.1
• Theorem B
• Corollary 1.2
• Corollary 1.3
• Lemma 2.1
• proof
• Corollary 2.2
• Lemma 2.3
• proof