Two classes of finite groups whose Chermak-Delgado lattice is a chain of length zero

TL;DR

This paper identifies two classes of finite groups where the Chermak-Delgado lattice is a single-element chain, expanding understanding of the structure of such groups.

Contribution

It characterizes two specific classes of finite groups with a Chermak-Delgado lattice of length zero, including groups formed by a normal abelian subgroup and a coprime abelian subgroup.

Findings

Chermak-Delgado lattice is a singleton for these groups

Groups with a normal abelian subgroup and coprime abelian subgroup have a specific lattice structure

The result generalizes previous findings on Chermak-Delgado lattices

Abstract

It is an open question in the study of Chermak-Delgado lattices precisely which finite groups $G$ have the property that $CD(G)$ is a chain of length $0$. In this note, we determine two classes of groups with this property. We prove that if $G=AB$ is a finite group, where $A$ and $B$ are abelian subgroups of relatively prime orders with $A$ normal in $G$, then the Chermak-Delgado lattice of $G$ equals $\{AC_B(A)\}$, a strengthening of earlier known results.