{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/23456"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/23456","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"A class of groups rich in finite quotients","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.","abstract_html":"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_has_math":false,"creators":["Walter, Vonn Andrew"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Robinson, Derek J.S."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T14:14:51Z","date_published":"2011-05-07T14:14:51Z","updated_at":"2026-07-22T22:25:22Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1994 Walter, Vonn Andrew"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9503344","(UMI)AAI9503344"],"render_values":[{"text":"AAI9503344","href":null,"code":true},{"text":"(UMI)AAI9503344","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/23456","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Robinson, Derek J.S."]},{"key":"dc:creator","label":"Author","values":["Walter, Vonn Andrew"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T14:14:51Z","10000-01-01","1994"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1994 Walter, Vonn Andrew"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9503344","(UMI)AAI9503344","http://hdl.handle.net/2142/23456"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Made available in DSpace on 2011-05-07T14:14:51Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9503344.pdf: 2321349 bytes, checksum: 9a7575b563e5508873b6958ea310c5eb (MD5) Previous issue date: 1994","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:04:36Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:30:53-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["A class of groups rich in finite quotients"]}]}],"canonical_facts":{"dc:contributor":["Robinson, Derek J.S."],"dc:creator":["Walter, Vonn Andrew"],"dc:date":["2011-05-07T14:14:51Z","10000-01-01","1994"],"dc: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.","Made available in DSpace on 2011-05-07T14:14:51Z (GMT). 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