Groups whose Chermak–Delgado lattice is a quasi-antichain

Abstract

Abstract A quasi-antichain is a lattice consisting of a maximum, a minimum, and the atoms of the lattice. The width of a quasi-antichain is the number of atoms. For a positive integer w ≥ 3 {w\geq 3} , a quasi-antichain of width w is denoted by ℳ w {\mathcal{M}_{w}} . In [B. Brewster, P. Hauck and E. Wilcox, Quasi-antichain Chermak–Delgado lattice of finite groups, Arch. Math. 103 2014, 4, 301–311], it is proved that ℳ w {\mathcal{M}_{w}} can be the Chermak–Delgado lattice of a finite group if and only if w = 1 + p a {w=1+p^{a}} for some positive integer a and some prime p. Let t be the number of abelian atoms in 𝒞 ⁢ 𝒟 ⁢ ( G ) {\mathcal{CD}(G)} . In this paper, we completely answer the following question: which values of t are possible in quasi-antichain Chermak–Delgado lattices?