Polycyclic groups, finite images, and elementary equivalence
type: Item
instance of: scholarly article
title: Polycyclic groups, finite images, and elementary equivalence
zbMATH Open document ID: 0702.20001
DOI: 10.1007/BF01196851
author: Gabriel Sabbagh
author: John S. Wilson
published in: Archiv der Mathematik
publication date: 1991-01-01T00:00:00Z
review text: This paper settles the following question: If G is a polycyclic group and H is a finitely generated group which is elementarily equivalent to G, must H be polycyclic? An affirmative answer follows from the paper's main result, which is purely algebraic in character. It asserts that, if G is polycyclic and H is a finitely generated residually finite group having exactly the same finite images as G, then H is polycyclic. The proof starts with a study of certain abelian groups and reduces the general case to the metabelian case. \(\forall \exists\) sentences provide the link between logic and algebra.
Mathematics Subject Classification ID: 20A15
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 20F16
zbMATH DE Number: 4150470
zbMATH Keywords: elementary equivalence
zbMATH Keywords: polycyclic group
zbMATH Keywords: finitely generated group
zbMATH Keywords: finitely generated residually finite group
zbMATH Keywords: finite images
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01196851
OpenAlex ID: W2024762423
description: scientific article; zbMATH DE number 4150470