Some Groups with Computable Chermak-Delgado Lattices

TL;DR

This paper investigates the structure of subgroups with maximal Chermak-Delgado measure in finite groups, showing they are subnormal, and explores their behavior in direct and wreath product constructions.

Contribution

It characterizes the Chermak-Delgado lattice in direct and wreath products and analyzes the properties of centrally large subgroups within this lattice.

Findings

Members of CD(G) are subnormal in G

Determined the Chermak-Delgado lattice for direct products

Described the lattice for certain wreath products

Abstract

Let G be a finite group and let H be a subgroup of G. The Chermak-Delgado measure of H with respect to G is the product of the order of H with the order of the centralizer of H. Originally described by A. Chermak and A. Delgado, the collection of all subgroups of G with maximal Chermak-Delgado measure, denoted CD(G), is a sublattice of the lattice of all subgroups of G. In this paper we note that the members of CD(G) are subnormal in G and determine the Chermak-Delgado lattice of direct products. We additionally describe the lattice for certain kinds of wreath products and examine the behavior of centrally large subgroups, a subset of the Chermak-Delgado lattice.