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 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.The Chermak-Delgado measure was originally defined by A. Chermak and A. Delgado as one in a family of functions from the subgroup lattice of a finite group into the positive integers. I. Martin Isaacs re-examined one of these function, dubbed it the Chermak-Delgado measure, and proved that subgroups with maximal Chermak-Delgado measure form a sublattice in the subgroup lattice of the group. B. Brewster and E. Wilcox then demonstrated that, for a direct product, this Chermak-Delgado lattice decomposes as the direct product of the Chermak-Delgado lattices of the factors, giving rise to the attention on the Chermak-Delgado lattice of p-groups (for a prime p) in this paper and others.The variety seen in the structure of the Chermak-Delgado lattices of p-groups seems inexhaustible; for example, there are many p-groups with a Chermak-Delgado lattice that is a single subgroup, a chain of arbitrary length, or a quasiantichain of width p + 1. In this paper we show that for any non-abelian p-group N with N is in its own Chermak-Delgado measure and Φ(N ) ≤ Z(N ), there exist two p-groups LE(m, n) and QE(n) with similar properties and such that the Chermak-Delgado lattices of LE(m, n) and QE(n) are the Chermak-Delgado lattice of N with either a m-diamond or a quasiantichain of width p + 1 (respectively) adjoined at both the maximum and minimum subgroups in the Chermak-Delgado lattice of N .

No abstract

No abstract