Über minimale k-irreguläre p-Gruppen. II. (On minimally k-irregular p-groups. II)
type: Item
instance of: scholarly article
title: Über minimale k-irreguläre p-Gruppen. II. (On minimally k-irregular p-groups. II)
zbMATH Open document ID: 0619.20005
author: Wolfgang Bannuscher
published in: Beiträge zur Algebra und Geometrie
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/138377
review text: This paper is a continuation of the author's previous work [Part I, ibid. 24, 111-123 (1987; Zbl 0618.20014)] on minimally k-irregular p-groups, and begins with two generalizations of criteria of Mann for regularity. Next a p-group is defined to be k-V-regular if all the finite groups in the variety generated by it are k-regular, and criteria for this are given. Conditions for a p-group to be \(p^ k\)-Abelian are then given, and it is shown that the variety generated by any minimally k-irregular p-group can be generated by a cyclic and a minimally k-irregular p-group of exponent \(p^{k+1}\). Next p-groups which are minimally k-irregular in the sense of subgroups are studied and generalizations of Mann's theorems are given. Finally it is shown that if P is a minimally k-irregular p- group of class \(c\geq p\) and exponent \(p^ n\) \((n>k)\), then there exists a p-group M minimally k-irregular in the sense of subgroups with the properties that M is of exponent \(p^ n\) and class c, the exponent of M' is at most \(p^ k\), P is a homomorphic image of M and M is isomorphic to a subgroup of the direct product of three certain quotient groups of it.
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20E10
zbMATH DE Number: 4004447
zbMATH Keywords: k-V-regular finite groups
zbMATH Keywords: generated by p-groups
zbMATH Keywords: minimally k-irregular p-groups
zbMATH Keywords: \(p^ k\)-Abelian
zbMATH Keywords: class
zbMATH Keywords: exponent
reviewed by: Norman Blackburn
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4004447