Finite groups with dense solitary subgroups

TL;DR

This paper characterizes all finite groups in which every non-empty open interval in their subgroup lattice contains a solitary subgroup, revealing a specific structural property of these groups.

Contribution

It provides a complete classification of finite groups with dense solitary subgroups in their subgroup lattice.

Findings

Identifies all finite groups with the property

Establishes the structural conditions for such groups

Contributes to the understanding of subgroup lattice properties

Abstract

A group $G$ is said to have dense solitary subgroups if each non-empty open interval in its subgroup lattice $L(G)$ contains a solitary subgroup. In this short note, we find all finite groups satisfying this property.