Chermak–Delgado Lattice Extension Theorems

Abstract

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 .

Groups whose Chermak–Delgado lattice is a quasi-antichain

“…In [2], it is proved that, for any integer n, a chain of length n can be a Chermak-Delgado lattice of a finite p-group. In [1], general conclusions are given.…”

“…Theorem 2. ( [1]) If L is a Chermak-Delgado lattice of a finite p-group G such that both G/Z(G) and G ′ are elementary abelian, then are L + and L ++ , where L + is a mixed 3-string with center component isomorphic to L and the remaining components being m-diamonds (a lattice with subgroups in the configuration of an m-dimensional cube), L ++ is a mixed 3-string with center component isomorphic to L and the remaining components being lattice isomorphic to M p+1 (a quasiantichain of width p + 1, see the following definition).…”

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

“…In the last years there has been a growing interest in understanding this lattice, especially for p-groups (see e.g. [1,2,3,10]). The study can be naturally extended to nilpotent groups, since by [2] the Chermak-Delgado lattice of a direct product of finite groups decomposes as the direct product of the Chermak-Delgado lattices of the factors.…”

Two classes of finite groups whose Chermak-Delgado lattice is a chain of length zero

“…In the last years there has been a growing interest in understanding this lattice (see e.g. [1,2,3,6,10,14]).…”