Finite groups with large Chermak-Delgado lattices

TL;DR

This paper characterizes finite groups whose Chermak-Delgado lattice size is just one or two less than their subgroup lattice size, revealing structural insights into these groups.

Contribution

It provides a complete classification of finite groups with Chermak-Delgado lattices nearly as large as their subgroup lattices, specifically for the cases where the difference is 1 or 2.

Findings

Identifies all finite groups with |CD(G)| = |L(G)| - 1

Identifies all finite groups with |CD(G)| = |L(G)| - 2

Establishes structural properties of these groups

Abstract

Given a finite group $G$, we denote by $L(G)$ the subgroup lattice of $G$ and by ${\cal CD}(G)$ the Chermak-Delgado lattice of $G$. In this note, we determine the finite groups $G$ such that $|{\cal CD}(G)|=|L(G)|-k$, $k=1,2$.