Finite groups with many isolated subgroups

TL;DR

This paper characterizes finite groups based on the number of isolated subgroups relative to their subgroup lattice size, focusing on cases where the difference is 0, 1, or 2.

Contribution

It provides a classification of finite groups with a specific small difference between the number of isolated subgroups and total subgroups.

Findings

Characterization of groups with no isolated subgroups

Identification of groups with exactly one or two isolated subgroups

Insights into the structure of subgroup lattices in finite groups

Abstract

Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\rm Isolated}(G)$ the set of isolated subgroups of $G$. In this note, we describe finite groups $G$ such that $|{\rm Isolated}(G)|=|L(G)|-k$, where $k=0,1,2$.