In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).

Finite Group Theory

Need a project to satisfy your craving for abstract algebra? Or do you have a student nosing around your office looking for a project? The Chermak–Delgado lattice is a sublattice of the subgroup la...

Exploring the Chermak-Delgado Lattice

In this note, we describe the structure of finite groups G whose Chermak–Delgado lattice is the interval [G∕Z(G)] = {H∈L(G)∣Z(G)≤H≤G}.

A note on the Chermak–Delgado lattice of a finite group

In this note we prove that the Chermak–Delgado lattice of a ZM-group is a chain of length 0, more precisely $${\mathcal {CD}}(\mathrm{ZM}(m,n,r))=\{H_{(1,d,0)}\}$$
 , where d is the multiplicative order of r modulo m. A similar conclusion is obtained for all dihedral groups $$D_{2m}$$
 with $$m\ne 4$$
 .

The Chermak–Delgado Lattice of ZM-Groups

The Chermak-Delgado lattice of a finite group is a modular, self-dual sublattice of the lattice of subgroups of $G$. The least element of the Chermak-Delgado lattice of $G$ is known as the Chermak-Delgado subgroup of $G$. This paper concerns groups with a trivial Chermak-Delgado subgroup. We prove that if the Chermak-Delgado lattice of such a group is lattice isomorphic to a Cartesian product of lattices, then the group splits as a direct product, with the Chermak-Delgado lattice of each direct factor being lattice isomorphic to one of the lattices in the Cartesian product. We establish many properties of such groups and properties of subgroups in the Chermak-Delgado lattice. We define a CD-minimal group to be an indecomposable group with a trivial Chermak-Delgado subgroup. We establish lattice theoretic properties of Chermak-Delgado lattices of CD-minimal groups. We prove an extension theorem for CD-minimal groups, and use the theorem to produce twelve examples of CD-minimal groups, each having different CD lattices. Curiously, quasi-antichain $p$-group lattices play a major role in the author's constructions.

Finite groups with a trivial Chermak–Delgado subgroup

If G is a finite group with subgroup H, then the Chermak–Delgado measure of H (in G) is defined as |H||C G (H)|. The Chermak–Delgado lattice of G, denoted &#x1d49e;&#x1d49f;(G), is the set of all subgroups with maximal Chermak–Delgado measure; this set is a moduar sublattice within the subgroup lattice of G. In this paper we provide an example of a p-group P, for any prime p, where &#x1d49e;&#x1d49f;(P) is lattice isomorphic to 2 copies of ℳ2 (a quasiantichain of width 2) that are adjoined maximum-to-minimum. We introduce terminology to describe this structure, called a 2-string of 2-diamonds, and we also give two constructions for generalizing the example. The first generalization results in a p-group with Chermak–Delgado lattice that, for any positive integers n and l, is a 2l-string of n-dimensional cubes adjoined maximum-to-minimum and the second generalization gives a construction for a p-group with Chermak–Delgado lattice that is a 2l-string of ℳ p+1 (quasiantichains, each of width p + 1) adjoined maximum-to-minimum.

Chermak–Delgado Lattice Extension Theorems

In a finite group G with subgroup H, the Chermak-Delgado measure of H (in G) is defined as the product of the order of H with the order of the centralizer of H. The Chermak-Delgado lattice of G, denoted CD(G), is the set of all subgroups with maximal Chermak-Delgado measure; this set is a sublattice within the subgroup lattice of G. In this paper we provide an example of a p-group P, for any prime p, where CD(P) is lattice isomorphic to 2 copies of M_4 (a quasiantichain of width 2) that are adjoined maximum-to-minimum. We introduce terminology to describe this structure, called a 2-string of 2-diamonds, and we also give two constructions for generalizing the example. The first generalization results in a p-group with Chermak-Delgado lattice that, for any positive integers n and l, is a 2l-string of n-dimensional cubes adjoined maximum-to-minimum and the second generalization gives a construction for a p-group with Chermak-Delgado lattice that is a 2l-string of M_(p+3) (quasiantichains, each of width p + 1) adjoined maximum-to-minimum.

Lijian An

Papers

Chermak–Delgado Lattice Extension Theorems

Chermak-Delgado Lattice Extension Theorems