A classification of 2-generator \(p\)-groups \(p\geq 3\) with many subgroups 2-subnormal
type: Item
instance of: scholarly article
title: A classification of 2-generator \(p\)-groups \(p\geq 3\) with many subgroups 2-subnormal
zbMATH Open document ID: 0534.20024
DOI: 10.1007/BF01193905
author: S. K. Mahdavianary
published in: Archiv der Mathematik
publication date: 1984-01-01T00:00:00Z
review text: In this paper, using results of C. Hobby, we show that if \(G\) is a 2-generator \(p\)-group, \(p\geq 3\), in which every cyclic subgroup is 2-subnormal, then in G the normalizer of every subgroup is a normal subgroup and \(G\) is a homomorphic image of \(G(p,r)\), where (i) If \(p>3\), then \(G(p,r)=<x,y| [x,y,x]=x^{p^ r}\), \([x,y,y]=y^{p^ r}\), \(x^{p^{2r}}=y^{p^{2r}}=1\), \(G_ 4=1>\), \(r>1;\) (ii) \(G(3,1)=<x,y| [x,y,x]=x^ 3=[x,y,y]=y^ 3\), \(y^ 9=1\), \(G_ 4=1>;\) (iii) If \(r>1\), then \(G(3,r)=<x,y| [x,y,x]=x^{3^ r}=[x,y,y]=y^{3^ r}\), \(y^{3^{2r}}=x^{3^{2r-1}}=1\), \(G_ 4=1>\).
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 3847662
zbMATH Keywords: 2-subnormal cyclic subgroup
zbMATH Keywords: 2-generator \(p\)-group
zbMATH Keywords: normalizer
MaRDI profile type: Publication
cites work: A class of three-Engel groups
cites work: Finite Groups with Normal Normalizers
cites work: On the H p -Problem for Finite p-Groups
cites work: A special class of three Engel groups
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description: scientific article; zbMATH DE number 3847662