A generalization of the Chermak-Delgado measure on subgroups and its associated lattice
type: Item
instance of: scholarly article
title: A generalization of the Chermak-Delgado measure on subgroups and its associated lattice
author: W. Cocke
author: Luise-Charlotte Kappe
author: Arturo Magidin
published in: International Journal of Group Theory
publication date: 2024-10-16T00:00:00Z
review text: Let \(G\) be a group acting on itself by conjugation. In [\textit{A. Chermak} and \textit{A. Delgado}, Proc. Am. Math. Soc. 107, No. 4, 907--914 (1989; Zbl 0687.20022)], for every \(r \in\mathbb{R}\), \(r>0\), a measure \(\mu_{r}\) that associates each subgroup \(H \leq G\) with a real number \(\mu_{r}(H)=|H|^{r}|C_{G}(H)|\) was introduced. If \(m_{r}=\max \{ \mu_{r}(H) \mid H \leq G \}\), then the set \(M_{r}(G)=\{ H \leq G \mid \mu_{r}(H)=m_{r} \}\) is a sublattice of the lattice \(\mathrm{Sub}(G)\) of all subgroups of \(G\).\N\NIn the paper under review, the authors consider mappings \(\mathcal{N}: \mathrm{Sub}(G)\rightarrow \mathrm{Sub}(G)\) and define a measure associated to \(\mathcal{N}\) by setting \(\nu(H)=|H||\mathcal{N}(H)|\). They investigate under what conditions on \(\mathcal{N}\) the subgroups with maximal measure form a sublattice of \(\mathrm{Sub}(G)\). In particular, their focus is on the case where \(\mathcal{N}(H)\) is a centralizer-like subgroup.
reviewed by: Egle Bettio
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20F22
zbMATH DE Number: 7927531
zbMATH Keywords: subgroup lattice
zbMATH Keywords: characteristic subgroup
zbMATH Keywords: Chermak-Delgado lattice
zbMATH Keywords: CD-admissible function
MaRDI profile type: Publication
cites work: Groups whose Chermak-Delgado lattice is a quasi-antichain
DOI: 10.22108/IJGT.2023.138469.1859
description: scientific article; zbMATH DE number 7927531