Groups of prime power order with derived subgroup of prime order
type: Item
instance of: scholarly article
title: Groups of prime power order with derived subgroup of prime order
zbMATH Open document ID: 0947.20014
author: Simon R. Blackburn
published in: Journal of Algebra
publication date: 1999-11-14T00:00:00Z
review text: A parametrization of groups of order \(p^n\) (\(p\) prime, \(n\geq 3\)) with commutator subgroup of order \(p\) is given. The set of parameters is independent of \(p\), although the proof for \(p=2\) is different from the odd case. Along the way a parametrization of the orbits of the general symplectic group on the set of flags of the underlying vector space is also obtained. Slightly reformulated, the main result establishes a bijection between the isomorphism classes of these groups and the following systems of nonnegative integers \((c,m_i,e,\alpha_{ij})\) such that \(c<n\), \(n-c\) is even, \(\sum im_i=c\), \(m_e>0\), \(\sum_{1\leq i\leq j\leq c+2}\alpha_{ij}=(n-c)/2\), and for all \(k\), \(2\leq k\leq c+2\) the sum \(\sum_{1\leq i\leq k}\alpha_{ik}+\sum_{k\leq j\leq c+2}\alpha_{kj}\) is at most \(m_{k-1}\), if \(k\leq e\); \(1\), if \(k=e+1\); \(m_e-1\), if \(k=e+2\); \(m_{k-2}\), if \(k>e+2\). The defining relations of the group corresponding to this system of parameters is too lengthy to be reproduced here (see Proposition 9 in the paper). We just note here that \(|Z(G)|=p^c\), \(m_i\) is the number of direct factors of order \(p^i\) in the canonical decomposition of \(Z(G)\), \(e\) is the largest integer such that \(G'\subseteq Z(G)^{p^{e-1}}\), and the \(\alpha_{ij}\) give the parametrization of a flag of \(G/Z(G)\) considered as a symplectic space.
reviewed by: Péter P. Pálfy
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 1346323
zbMATH Keywords: finite \(p\)-groups
zbMATH Keywords: commutator subgroups
zbMATH Keywords: general symplectic groups
zbMATH Keywords: relations
Wikidata QID: Q60204646
MaRDI profile type: Publication
full work available at URL: jabr.1998.7909
OpenAlex ID: W1982149942
cites work: Q5577154
cites work: Enumerating p -Groups. I: Inequalities
cites work: Über den Frobenius'schen Klassenbegriff in nilpotenten Gruppen
cites work: Odd order nilpotent groups of class two with cyclic centre
cites work: Q4186538
cites work: What is the probability that two elements of a finite group commute?
DOI: 10.1006/JABR.1998.7909
description: scientific article; zbMATH DE number 1346323