A class of groups rich in finite quotients
type: CreativeWork
description: If X is a class of groups, the class of counter-X groups is defined to consist of all groups having no non-trivial X-quotients. Counter-counter-finite groups are studied here; any non-trivial quotient of such a group has a non-trivial representation over any finitely generated domain, so we shall call these groups highly representable or HR-groups. Abelian, nilpotent, and solvable HR-groups are examined in detail, with structure theorems given in the abelian and nilpotent cases. Investigation of a subclass of solvable HR-groups leads to a generalization of Gruenberg's Theorem on the residual finiteness of finitely generated torsion-free nilpotent groups. Additional topics include characterizations of the HR radical and residual in groups with finite composition length, as well as the normal and subnormal structure of HR-groups.
abstract: If X is a class of groups, the class of counter-X groups is defined to consist of all groups having no non-trivial X-quotients. Counter-counter-finite groups are studied here; any non-trivial quotient of such a group has a non-trivial representation over any finitely generated domain, so we shall call these groups highly representable or HR-groups. Abelian, nilpotent, and solvable HR-groups are examined in detail, with structure theorems given in the abelian and nilpotent cases. Investigation of a subclass of solvable HR-groups leads to a generalization of Gruenberg's Theorem on the residual finiteness of finitely generated torsion-free nilpotent groups. Additional topics include characterizations of the HR radical and residual in groups with finite composition length, as well as the normal and subnormal structure of HR-groups.
author: Walter, Vonn Andrew
language: eng
keywords: Mathematics
date published: 2011-05-07T14:14:51Z
date modified: 2026-07-22T22:25:22Z
provider: University of Illinois - Urbana-Champaign
publisher: Networked Digital Library of Theses and Dissertations
type: WebPage
source organization: University of Illinois at Urbana-Champaign

PropertyValue
property ID: Handle
value: http://hdl.handle.net/2142/23456