Nilpotent groups of class two that can appear as central quotient groups
type: Item
instance of: scholarly article
title: Nilpotent groups of class two that can appear as central quotient groups
zbMATH Open document ID: 0722.20011
author: Hermann Heineken
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1990-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108200
review text: Suppose that H is a finite group in which \(H'/H'\cap Z(H)\) is central and elementary Abelian of order \(p^ n\). Let \(| H/Z_ 2(H)| =p^ m\). It has long been known that if \(n=1\), then \(m=2\). In the present paper, it is shown that if \(n=2\), then \(2<m<6\), and examples are given which show that for p odd, these bounds are sharp. The proof involves the study of vector spaces with two linearly independent alternating forms. An example is given for general n in which \(m=n(n+3)\).
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20F14
Mathematics Subject Classification ID: 20F18
zbMATH DE Number: 4190008
zbMATH Keywords: p-groups
zbMATH Keywords: finite group
zbMATH Keywords: vector spaces
zbMATH Keywords: alternating forms
reviewed by: Norman Blackburn
MaRDI profile type: Publication
cites work: Group extensions, representations, and the Schur multiplicator
cites work: Paare alternierender Formen
cites work: Groups of class 2 and exponent 𝑝 with commutator subgroup of order 𝑝²
description: scientific article; zbMATH DE number 4190008