Groups of prime-power order having an Abelian centralizer of type (r,1)
type: Item
instance of: scholarly article
title: Groups of prime-power order having an Abelian centralizer of type (r,1)
zbMATH Open document ID: 0552.20010
DOI: 10.1007/BF01300734
author: Norman Blackburn
published in: Monatshefte für Mathematik
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/178209
review text: Let G be a group of order \(p^ n\), where p is an odd prime, having an element s of order p the centralizer L of which is the direct product of \(\) and a cyclic group \(<t>\) of order \(p^ r\). The aim of the paper is to examine the commutator and power structure of G (assuming \(n>r+2)\). The first assertion is that the class of G is n-r, and there is a unique subgroup of each possible order between L and G. Let \(G=G_ 1>G_ 2>...>G_{n-r}>G_{n-r+1}=1\) be the lower central series of G. Roughly speaking, \(| G_ i:G_{i+1}|\) is in general p for \(i\geq 3\), but there may be bulges in which \(| G_ i:G_{i+1}| =p^ 2.\) The first such bulge will occur with \(i=2h-1\), where \(2\leq h\leq (p+1),\) and there are at most r-2 bulges which occur for \(i=2h-1,\quad 2h-1+p- 1),...,2h-1+r'(p-1).\) Thus the lower central series of G has a tail of factors of order p, and this tail must have length at least p-1. As for the power structure, if \(s_ 1\) is any element of \(G-LG_ 2\) and \(s_ i=[s,s_{i-1}]\) \((i=2,...,n-r)\), then \(s_ i\in G_ i-G_{i+1}\) and \(s^ p_ is_{i+p-1}\in G_{i+p}\) \((i=2,...,n-p-r+1)\). Thus the group generated by the p-th powers of the elements of \(G_ i\) is \(G_{i+p-1}\) for \(i=3,...,n-r-p+2\).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20F12
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 3879132
zbMATH Keywords: centralizer
zbMATH Keywords: commutator
zbMATH Keywords: class
zbMATH Keywords: lower central series
zbMATH Keywords: group generated by the p-th powers
MaRDI profile type: Publication
cites work: On a special class of \(p\)-groups
cites work: On Centralizers in P -Groups
cites work: A Contribution to the Theory of Groups of Prime-Power Order
cites work: Endliche Gruppen I
cites work: ON p-GROUPS OF MAXIMAL CLASS I
OpenAlex ID: W2006322031
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description: scientific article; zbMATH DE number 3879132