Groups whose Chermak-Delgado lattice is a quasi-antichain

TL;DR

This paper investigates the structure of groups whose Chermak-Delgado lattice forms a quasi-antichain, establishing a relationship between parameters defining the lattice and the group's properties.

Contribution

It proves that for such groups, the parameters a and b, related to the lattice's width and abelian atoms, satisfy a=b or a=2b, advancing understanding of their structure.

Findings

Established the condition a=b or a=2b for groups with Chermak-Delgado quasi-antichain lattices.

Extended the characterization of Chermak-Delgado lattices in finite groups.

Connected lattice parameters to group-theoretic properties.

Abstract

A quasiantichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasiantichian is the number of atoms. For a positive integer $w$ ($\ge 3$), a quasiantichain of width $w$ is denoted by $\mathcal{M}_{w}$. In \cite{BHW2}, it is proved that $\mathcal{M}_{w}$ can be as a Chermak-Delgado lattice of a finite group if and only if $w=1+p^a$ for some positive integer $a$. Let $t$ be the number of abelian atoms in $\mathcal{CD}(G)$. If $t>2$, then, according to \cite{BHW2}, there exists a positive integer $b$ such that $t=p^b+1$. The converse is still an open question. In this paper, we proved that $a=b$ or $a=2b$.