{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/21391"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/21391","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"On an analogue to the Grushko-Neumann theorem involving finite groups","abstract":"Let C be a class of finite groups closed under the operations of taking subgroups, quotients, and extensions. Let H and K be pro-C-groups and let $G=H*K$ be their free pro-C-product. An open question in the theory of profinite groups is whether or not $d(G)=d(H)+d(K),$ which is an analogue to the Grushko-Neumann theorem for abstract free products of groups.","abstract_html":"Let C be a class of finite groups closed under the operations of taking subgroups, quotients, and extensions. Let H and K be pro-C-groups and let $G=H*K$ be their free pro-C-product. An open question in the theory of profinite groups is whether or not $d(G)=d(H)+d(K),$ which is an analogue to the Grushko-Neumann theorem for abstract free products of groups.","abstract_has_math":true,"creators":["Dabrowski, Walter Casimir"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Rotman, Joseph J."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:07:19Z","date_published":"2011-05-07T13:07:19Z","updated_at":"2026-07-22T22:25:17Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1996 Dabrowski, Walter Casimir"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["9780591086799","AAI9702494","(UMI)AAI9702494"],"render_values":[{"text":"9780591086799","href":null,"code":true},{"text":"AAI9702494","href":null,"code":true},{"text":"(UMI)AAI9702494","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/21391","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Rotman, Joseph J."]},{"key":"dc:creator","label":"Author","values":["Dabrowski, Walter Casimir"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:07:19Z","10000-01-01","1996"]},{"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 1996 Dabrowski, Walter Casimir"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["9780591086799","AAI9702494","(UMI)AAI9702494","http://hdl.handle.net/2142/21391"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let C be a class of finite groups closed under the operations of taking subgroups, quotients, and extensions. Let H and K be pro-C-groups and let $G=H*K$ be their free pro-C-product. An open question in the theory of profinite groups is whether or not $d(G)=d(H)+d(K),$ which is an analogue to the Grushko-Neumann theorem for abstract free products of groups.","Ribes and Wong show that the two following assertions are equivalent: (1) For every pair H and K of pro-C-groups, $d(H\\ *\\ K)=d(H)+d(K).$ (2) For every pair of H and K of finite groups in C, there is a finite group G in C such that $G={},\\ {\\rm and}\\ d(G)=d(H)+d(K).$","Using the classification of finite simple groups, we provide evidence that assertion (2) in the case that C consists of all finite groups is false for certain H and K. This might be thought of as an extension of the evidence provided so far by Kovacs, Sim, and Lucchini.","Made available in DSpace on 2011-05-07T13:07:19Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9702494.pdf: 3276222 bytes, checksum: 7598234d70ede8407d00980d0b1bd3e0 (MD5) Previous issue date: 1996","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:50:27Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:23:03-05:00 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":["On an analogue to the Grushko-Neumann theorem involving finite groups"]}]}],"canonical_facts":{"dc:contributor":["Rotman, Joseph J."],"dc:creator":["Dabrowski, Walter Casimir"],"dc:date":["2011-05-07T13:07:19Z","10000-01-01","1996"],"dc:description":["Let C be a class of finite groups closed under the operations of taking subgroups, quotients, and extensions. Let H and K be pro-C-groups and let $G=H*K$ be their free pro-C-product. An open question in the theory of profinite groups is whether or not $d(G)=d(H)+d(K),$ which is an analogue to the Grushko-Neumann theorem for abstract free products of groups.","Ribes and Wong show that the two following assertions are equivalent: (1) For every pair H and K of pro-C-groups, $d(H\\ *\\ K)=d(H)+d(K).$ (2) For every pair of H and K of finite groups in C, there is a finite group G in C such that $G={},\\ {\\rm and}\\ d(G)=d(H)+d(K).$","Using the classification of finite simple groups, we provide evidence that assertion (2) in the case that C consists of all finite groups is false for certain H and K. This might be thought of as an extension of the evidence provided so far by Kovacs, Sim, and Lucchini.","Made available in DSpace on 2011-05-07T13:07:19Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9702494.pdf: 3276222 bytes, checksum: 7598234d70ede8407d00980d0b1bd3e0 (MD5) Previous issue date: 1996","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:50:27Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:23:03-05:00 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"],"dc:identifier":["9780591086799","AAI9702494","(UMI)AAI9702494","http://hdl.handle.net/2142/21391"],"dc:language":["eng"],"dc:rights":["Copyright 1996 Dabrowski, Walter Casimir"],"dc:subject":["Mathematics"],"dc:title":["On an analogue to the Grushko-Neumann theorem involving finite groups"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:17Z"}