{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/86777"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/86777","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Some Results on Free Groups in Combinatorial Group Theory","abstract":"Let Fm be a free group of a finite rank m ≥ 2. We prove that there exist two elements u 1, u2 ∈ Fm such that every endomorphism y of Fm with non-cyclic image is completely determined by y (u1), y (u2). We obtain this result as a corollary of the construction of a C-test word vn( x1,...,xn), for each n ≥ 2, with the additional property that vn (X1,...,Xn) = 1 if and only if the subgroup ⟨X1,..., Xn⟩ of Fm generated by X1,...,Xn is cyclic. We also prove that every primitivity preserving endomorphism of Fm with m ≥ 3 is an automorphism.","abstract_html":"Let Fm be a free group of a finite rank m ≥ 2. We prove that there exist two elements u 1, u2 ∈ Fm such that every endomorphism y of Fm with non-cyclic image is completely determined by y (u1), y (u2). We obtain this result as a corollary of the construction of a C-test word vn( x1,...,xn), for each n ≥ 2, with the additional property that vn (X1,...,Xn) = 1 if and only if the subgroup ⟨X1,..., Xn⟩ of Fm generated by X1,...,Xn is cyclic. We also prove that every primitivity preserving endomorphism of Fm with m ≥ 3 is an automorphism.","abstract_has_math":false,"creators":["Lee, Donghi"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Sergei v. Ivanov"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-09-28T15:19:29Z","date_published":"2015-09-28T15:19:29Z","updated_at":"2026-07-22T22:26:27Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(MiAaPQ)AAI3017139"],"render_values":[{"text":"(MiAaPQ)AAI3017139","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/86777","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Sergei v. 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We prove that there exist two elements u 1, u2 ∈ Fm such that every endomorphism y of Fm with non-cyclic image is completely determined by y (u1), y (u2). We obtain this result as a corollary of the construction of a C-test word vn( x1,...,xn), for each n ≥ 2, with the additional property that vn (X1,...,Xn) = 1 if and only if the subgroup ⟨X1,..., Xn⟩ of Fm generated by X1,...,Xn is cyclic. We also prove that every primitivity preserving endomorphism of Fm with m ≥ 3 is an automorphism.","Made available in DSpace on 2015-09-28T15:19:29Z (GMT). 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We prove that there exist two elements u 1, u2 ∈ Fm such that every endomorphism y of Fm with non-cyclic image is completely determined by y (u1), y (u2). We obtain this result as a corollary of the construction of a C-test word vn( x1,...,xn), for each n ≥ 2, with the additional property that vn (X1,...,Xn) = 1 if and only if the subgroup ⟨X1,..., Xn⟩ of Fm generated by X1,...,Xn is cyclic. We also prove that every primitivity preserving endomorphism of Fm with m ≥ 3 is an automorphism.","Made available in DSpace on 2015-09-28T15:19:29Z (GMT). No. of bitstreams: 2 license.txt: 4848 bytes, checksum: 96035ab3f5e1c23cc7138a224ce498bd (MD5) 3017139.pdf: 2719348 bytes, checksum: a2785a0933fd3c99f1fd7d1aa657cd19 (MD5) Previous issue date: 2001","Embargo set by: Seth Robbins for item 88058 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","67 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2001."],"dc:identifier":["http://hdl.handle.net/2142/86777","(MiAaPQ)AAI3017139"],"dc:language":["eng"],"dc:subject":["Mathematics"],"dc:title":["Some Results on Free Groups in Combinatorial Group Theory"],"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":"2026-07-22T22:26:27Z"}