Quasi-Antichain Chermak-Delgado Lattices of Finite Groups

TL;DR

This paper investigates the structure of Chermak-Delgado lattices in finite groups, proving that quasi-antichain intervals relate to elementary abelian p-groups and exploring their properties and examples.

Contribution

It characterizes the structure of quasi-antichain intervals in Chermak-Delgado lattices and constructs examples of such groups, revealing their connection to elementary abelian p-groups.

Findings

Quasi-antichain intervals correspond to elementary abelian p-groups.

Number of atoms in the quasi-antichain relates to a prime p.

Examples of groups with quasi-antichain Chermak-Delgado lattices are provided.

Abstract

The Chermak-Delgado lattice of a finite group is a dual, modular sublattice of the subgroup lattice of the group. This paper considers groups with a quasi-antichain interval in the Chermak-Delgado lattice, ultimately proving that if there is a quasi-antichain interval between $L$ and $H$ with $L \leq H$ then there exists a prime $p$ such that the quotient $H / L$ is an elementary abelian $p$-group and the number of atoms in the quasi-antichain is one more than a power of $p$. In the case where the Chermak-Delgado lattice of the entire group is a quasi-antichain, the relationship between the number of abelian atoms and the prime $p$ is examined; additionally several examples of group with a quasi-antichain Chermak-Delgado lattice are constructed.