Soluble products of minimax groups, and nearly surjective derivations
type: Item
instance of: scholarly article
title: Soluble products of minimax groups, and nearly surjective derivations
zbMATH Open document ID: 0654.20025
DOI: 10.1016/0022-4049(88)90129-6
author: John S. Wilson
published in: Journal of Pure and Applied Algebra
publication date: 1988-01-01T00:00:00Z
review text: The author proves that a soluble group \(G=AB\) which is the product of two minimax subgroups \(A\) and \(B\) is a minimax group. (A soluble group is a minimax group if it has a finite series whose factors are cyclic or Prüfer groups.) This solves Problem 9.54 of the Kourouvka Notebook [1986; Zbl 0625.20001]. Some special cases of this result where one of the two subgroups \(A\) or \(B\) satisfies a nilpotency requirement, were previously dealt with by D. I. Zaicev, D. Robinson and the reviewer. For the proof of the above theorem one needs to show that whenever \(\beta\) is a (nearly) surjective derivation from a soluble minimax group \(H\) to an \(H\)-module \(M\), then, regarded as a group, \(M\) is minimax. (Such a derivation is called nearly surjective if the set \(H\beta\) has a finite complement in \(M\).) After some reductions one has to prove that there is no simple module \(M\) for a soluble minimax group \(H\) with the property that there is a (nearly) surjective derivation from \(H\) to \(M\).
Mathematics Subject Classification ID: 20E22
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20F16
Mathematics Subject Classification ID: 20F19
zbMATH DE Number: 4067193
zbMATH Keywords: factorized groups
zbMATH Keywords: soluble groups
zbMATH Keywords: product of minimax subgroups
zbMATH Keywords: minimax groups
zbMATH Keywords: surjective derivations
zbMATH Keywords: soluble minimax groups
reviewed by: Bernhard Amberg
MaRDI profile type: Publication
full work available at URL: 0022-4049(88)90129-6
OpenAlex ID: W2053574115
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description: scientific article; zbMATH DE number 4067193