Classification of 2-generator finite metabelian p-groups having a proper \(H_ p\)-subgroup
type: Item
instance of: scholarly article
title: Classification of 2-generator finite metabelian p-groups having a proper \(H_ p\)-subgroup
zbMATH Open document ID: 0633.20011
DOI: 10.1007/BF01191262
author: William F. Cody
published in: Archiv der Mathematik
publication date: 1989-01-01T00:00:00Z
review text: For a group G and prime p, the subgroup of G generated by the elements of order unequal to p is called the \(H_ p\)-subgroup. Hughes conjectured that whenever \(H_ p(G)\) is a proper subgroup, it must have index p. Although in general false, the Hughes conjecture has been verified in many important classes of groups. In particular, Hogan and Kappe established its truth for finite metabelian p-groups. Additionally, they constructed an example of a 2-generator finite metabelian p-group of class p having a proper abelian \(H_ p\)-subgroup. In this paper, we generalize that construction to exhibit a family of 2-generator finite metabelian p-groups of class m, \(m\geq p\), each having a proper \(H_ p\)- subgroup of class n, \(1\leq n\leq p-2\). These values of the ordered pairs (m,n) span the full range of possibilities. We then prove that any 2- generator finite metabelian p-group having a proper \(H_ p\)-subgroup must be a homomorphic image of a group in this family. That these groups, composed ``mostly'' of elements of order p, can be constructed of arbitrary size is in contrast to the situation for finite 2-generator groups of exponent p described in a well-known result of Kostrikin. Finally, the study of these 2-generator groups enables us to obtain some interesting structural theorems for the general n-generator case.
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 4029812
zbMATH Keywords: Hughes conjecture
zbMATH Keywords: \(H_ p\)-subgroup
zbMATH Keywords: 2-generator finite metabelian p- groups
zbMATH Keywords: Hughes subgroup
MaRDI profile type: Publication
cites work: A Meier-Wunderli theorem for \(H_ p\)-subgroups
cites work: On the Hughes conjecture
cites work: More on the Hughes conjecture
cites work: Q3253828
cites work: On the H p -Problem for Finite p-Groups
cites work: Partial Difference Sets
cites work: Generalized incidence matrices over group algebras
cites work: The \(H_p\)-problem and the structure of \(H_p\)-groups
cites work: Endliche Gruppen I
cites work: Die Nilpotenz der \(H_ p\)-Gruppen
cites work: Q3263741
cites work: Metabelsche Gruppen
cites work: Q5513799
description: scientific article; zbMATH DE number 4029812