Groups Whose Chermak-Delgado Lattice is a Chain

TL;DR

This paper investigates finite groups where the Chermak-Delgado lattice forms a chain, providing conditions for this structure and constructing specific p-groups with chain-like Chermak-Delgado lattices of any length.

Contribution

It introduces conditions for Chermak-Delgado lattices to be chains and constructs p-groups with arbitrary chain lengths based on these conditions.

Findings

Chermak-Delgado lattice can be a chain under certain conditions.

Constructed p-groups with Chermak-Delgado lattice as a chain of any length.

Provided a method to extend Chermak-Delgado lattices in finite groups.

Abstract

For a finite group G with subgroup H the Chermak-Delgado measure of H in G refer to the product of the order of H with the order of its centralizer, C_G(H). The set of all subgroups with maximal Chermak-Delgado measure form a sublattice, CD(G), within the subgroup lattice of G. This paper examines conditions under which the Chermak-Delgado lattice is a chain of subgroups. On the basis of a general result how to extend certain Chermak-Delgado lattices, we construct for any prime p and any non-negative integer n a p-group whose Chermak-Delgado lattice is a chain of length n.