On the Chermak-Delgado lattice of a finite group

TL;DR

This paper studies the Chermak-Delgado lattice ${\cal CD}(G)$ of finite groups under index constraints, with a focus on applying the results to dicyclic groups and metabelian $p$-groups of maximal class. It establishes a general classification (Theorem 2.1) that ties the lattice structure to the index $|G:Z(G)|$ and to the existence of a maximal self-centralizing subgroup, yielding chains, quasi-antichains, or more intricate intervals depending on parameter ranges. The authors then determine ${\cal CD}(G)$ for generalized dicyclic groups $Dic_{4n}(A)$ in terms of the abelian group $A$ (and its exponent), including a width-3 quasi-antichain case when $A$ has type $\mathbb{Z}_2^m\times\mathbb{Z}_4$, and a complete description for $Dic_{4n}$. Finally, they classify ${\cal CD}(G)$ for metabelian $p$-groups of maximal class, with detailed outcomes for small orders and for larger groups when a subgroup with $|T:Z(T)|=p^2$ exists, culminating in thorough case-by-case lattices and several structural corollaries. These results advance understanding of how index data governs CD lattices in key finite-group families and provide explicit lattice descriptions for concrete groups.

Abstract

By imposing conditions upon the index of a self-centralizing subgroup of a group, and upon the index of the center of the group, we are able to classify the Chermak-Delgado lattice of the group. This is our main result. We use this result to classify the Chermak-Delgado lattices of dicyclic groups and of metabelian $p$-groups of maximal class.

Theorems & Definitions (17)

• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Theorem 2.1
• proof
• Corollary 2.2
• Theorem 3.1
• Corollary 3.2
• Theorem 3.3
• Corollary 3.4