Groups whose Chermak-Delgado lattice is a quasi-antichain

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 $\mathcal{C}\mathcal{D}(G)$. 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 $a=2b$.