Chermak-Delgado Lattice Extension Theorems
Abstract
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.
Keywords
Cite
@article{arxiv.1307.0353,
title = {Chermak-Delgado Lattice Extension Theorems},
author = {Lijian An and Joseph Brennan and Haipeng Qu and Elizabeth Wilcox},
journal= {arXiv preprint arXiv:1307.0353},
year = {2014}
}
Comments
9 pages, 1 figure A small (but notable) error in the group order in Construction 1.2 has been fixed. This paper has been accepted by Communications in Algebra, to appear as of May 2014