A problem of Kegel
type: Item
instance of: scholarly article
title: A problem of Kegel
zbMATH Open document ID: 0809.20016
DOI: 10.1007/BF01262177
published in: Mathematical Notes
publication date: 1994-04-12T00:00:00Z
review text: Let \(\mathfrak X\) be a class of finite groups which is closed under subgroups, homomorphic images and extensions. A subgroup \(H\) of a finite group \(G\) is said to be \(\mathfrak X\)-normal in \(G\) if there exists a chain of subgroups \(H = H_ 0 \leq H_ 1 \leq \dots \leq H_ n = G\) such that either \(H_ i\) is normal in \(H_{i + 1}\) or \(H_{i + 1}/(H_ i)_{H_{i + 1}}\) belongs to \(\mathfrak X\) for every \(i < n\) (here \((H_ i)_{H_{i + 1}}\) denotes the core of \(H_ i\) in \(H_{i + 1}\)). Let \(G^{\mathfrak X}\) denote the \({\mathfrak X}\)-coradical of the finite group \(G\), i.e. the smallest normal subgroup of \(G\) such that \(G/G^{\mathfrak X}\) is an \(\mathfrak X\)-group. Answering a question of \textit{O. H. Kegel} [Arch. Math. 30, 225-228 (1978)], the author proves that if \(H\) is an \(\mathfrak X\)- normal subgroup of a finite group \(G\) and \(H = H^{\mathfrak X} = H'\), then \(HK = KH\) for every \(\mathfrak X\)-normal subgroup \(K\) of \(G\).
Mathematics Subject Classification ID: 20D35
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 525670
zbMATH Keywords: coradical
zbMATH Keywords: finite groups
zbMATH Keywords: chain of subgroups
zbMATH Keywords: core
zbMATH Keywords: \(\mathfrak X\)-normal subgroup
author: S. F. Kamornikov
MaRDI profile type: Publication
cites work: Subgroup lattices of finite groups which properly contain the lattice of subnormal subgroups
cites work: Q3033956
cites work: Q3961010
cites work: Über den Normalisator der subnormalen Untergruppen
description: scientific article; zbMATH DE number 525670