On nilpotent groups on exponent p
type: Item
instance of: scholarly article
title: On nilpotent groups on exponent p
zbMATH Open document ID: 0685.20031
DOI: 10.1016/0021-8693(89)90253-6
author: Berthold J. Maier
published in: Journal of Algebra
publication date: 1989-01-01T00:00:00Z
review text: The paper discusses amalgams of groups and applications to countable existentially closed groups. An amalgam of groups consists of a pair of groups A, B with a common subgroup D and a realization of this amalgam consists of a group C together with embeddings of A and B into C which co-incide on D. The general problem considered is to find such an amalgam with C suitable restricted relative to A and B. Let p be a prime. Let \({\mathbb{K}}\) denote the class of nilpotent groups of exponent p and class less than p. Let \({\mathbb{K}}'\) denote the class of nilpotent groups with central quotient of exponent p. Then criteria are given for an amalgam of groups in \({\mathbb{K}}\) or \({\mathbb{K}}'\) to be realizable in \({\mathbb{K}}\) or \({\mathbb{K}}'\). The author also describes the groups D in either of these two classes which have the property that any amalgam with common subgroup D can be realized within the class. The results are then applied to the existence and uniqueness of countable existentially closed groups within various classes. Amongst other things, it is shown that the class \({\mathbb{K}}\) has a unique countable existentially closed group and that the class of all torsion groups in \({\mathbb{K}}'\) has a unique countable existentially closed group.
Mathematics Subject Classification ID: 20F18
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 4122225
zbMATH Keywords: amalgams of groups
zbMATH Keywords: embeddings
zbMATH Keywords: nilpotent groups of exponent p
zbMATH Keywords: countable existentially closed groups
reviewed by: J. R. J. Groves
MaRDI profile type: Publication
full work available at URL: 0021-8693(89)90253-6
OpenAlex ID: W2048983567
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description: scientific article; zbMATH DE number 4122225