Chermak–Delgado Lattice Extension Theorems |
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Authors: | Lijian An Joseph Phillip Brennan Haipeng Qu |
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Institution: | 1. Department of Mathematics , Shanxi Normal University , Linfen , China;2. Department of Mathematical Sciences , Binghamton University , Binghamton , New York , USA |
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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 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 𝒞𝒟(P) is lattice isomorphic to 2 copies of ?2 (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 ? p+1 (quasiantichains, each of width p + 1) adjoined maximum-to-minimum. |
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Keywords: | Lattices of subgroups p-Groups |
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