A generalization of Kramer's theory
type: Item
instance of: scholarly article
title: A generalization of Kramer's theory
zbMATH Open document ID: 1449.20012
author: A. N. Skiba
author: Zhang Chi
published in: Acta Mathematica Hungarica
publication date: 2020-08-11T00:00:00Z
review text: Let \(\sigma\) be a partition of the set of all prime numbers, and for each positive integer \(n\) let \(\sigma(n)\) denote the set of all elements of \(\sigma\) that contain some divisor of \(n\). Then the positive integers \(m\) and \(n\) are called \textit{\(\sigma\)-coprime} if \(\sigma(m)\cap\sigma(n)=\emptyset\). A group class \(\mathcal{F}\) (consisting only of finite groups) is said to be \textit{\(\Gamma_t^\sigma\)-closed} for some integer \(t>1\) if \(\mathcal{F}\) contains every finite group \(G\) admitting \(t\) subgroups \(A_1,\ldots,A_t\) such that \(G=A_iA_j\) whenever \(i\neq j\) and the indices \(|G:N_G(A_1)|,\ldots,|G:N_G(A_t)|\) are pairwise \(\sigma\)-coprime.\par In the paper under review, the authors study the above concepts, and in particular they prove that certain formations of generalized metanilpotent groups (related to the partition \(\sigma\)) are \(\Gamma_4^\sigma\)-closed.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 7232864
zbMATH Keywords: finite group
zbMATH Keywords: formation \(\sigma\)-function
zbMATH Keywords: \(\sigma\)-local formation
zbMATH Keywords: (weakly) \(\Gamma_t^{\sigma}\)-closed class of groups
zbMATH Keywords: \(\sigma\)-soluble group
zbMATH Keywords: \(\sigma\)-nilpotent group
MaRDI profile type: Publication
full work available at URL: s10474-018-00902-5
OpenAlex ID: W2908411140
DOI: 10.1007/S10474-018-00902-5
description: scientific article; zbMATH DE number 7232864