{"id":"https://openalex.org/W2267943872","doi":"https://doi.org/10.1142/s0218196716500223","title":"Non-triviality of some one-relator products of three groups","display_name":"Non-triviality of some one-relator products of three groups","publication_year":2016,"publication_date":"2016-04-07","ids":{"openalex":"https://openalex.org/W2267943872","doi":"https://doi.org/10.1142/s0218196716500223","mag":"2267943872"},"language":"en","primary_location":{"id":"doi:10.1142/s0218196716500223","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218196716500223","pdf_url":null,"source":{"id":"https://openalex.org/S23260266","display_name":"International Journal of Algebra and Computation","issn_l":"0218-1967","issn":["0218-1967","1793-6500"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Algebra and Computation","raw_type":"journal-article"},"type":"preprint","indexed_in":["arxiv","crossref","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/1509.02717","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5066229157","display_name":"Ihechukwu Chinyere","orcid":"https://orcid.org/0000-0002-0071-8085"},"institutions":[{"id":"https://openalex.org/I98220444","display_name":"Michael Okpara University of Agriculture","ror":"https://ror.org/050850526","country_code":"NG","type":"education","lineage":["https://openalex.org/I98220444"]}],"countries":["NG"],"is_corresponding":true,"raw_author_name":"Ihechukwu Chinyere","raw_affiliation_strings":["Department of Mathematics, College of Physical and Applied Sciences, Michael Okpara University of Agriculture, Umudike P.M.B. 7267 Umuahia, Abia State, Nigeria"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, College of Physical and Applied Sciences, Michael Okpara University of Agriculture, Umudike P.M.B. 7267 Umuahia, Abia State, Nigeria","institution_ids":["https://openalex.org/I98220444"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5114018447","display_name":"James Howie","orcid":"https://orcid.org/0000-0001-9975-1936"},"institutions":[{"id":"https://openalex.org/I4210117017","display_name":"Maxwell Institute for Mathematical Sciences","ror":"https://ror.org/02tsqtg57","country_code":"GB","type":"facility","lineage":["https://openalex.org/I32062511","https://openalex.org/I4210117017","https://openalex.org/I98677209"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"James Howie","raw_affiliation_strings":["Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot\u2013Watt University, Edinburgh EH14 4AS, Scotland"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Maxwell Institute for Mathematical Sciences, Heriot\u2013Watt University, Edinburgh EH14 4AS, Scotland","institution_ids":["https://openalex.org/I4210117017"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5066229157"],"corresponding_institution_ids":["https://openalex.org/I98220444"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.03753364,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"26","issue":"03","first_page":"533","last_page":"550"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9991000294685364,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.9983999729156494,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/triviality","display_name":"Triviality","score":0.921796441078186},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.8133801817893982},{"id":"https://openalex.org/keywords/quotient","display_name":"Quotient","score":0.7756091356277466},{"id":"https://openalex.org/keywords/closure","display_name":"Closure (psychology)","score":0.6261361837387085},{"id":"https://openalex.org/keywords/free-product","display_name":"Free product","score":0.6056808233261108},{"id":"https://openalex.org/keywords/product","display_name":"Product (mathematics)","score":0.59068763256073},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.5234445929527283},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.5117749571800232},{"id":"https://openalex.org/keywords/element","display_name":"Element (criminal law)","score":0.44347333908081055},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.3201841115951538},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.1034184992313385},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.05841827392578125},{"id":"https://openalex.org/keywords/economics","display_name":"Economics","score":0.03694123029708862},{"id":"https://openalex.org/keywords/law","display_name":"Law","score":0.032551735639572144}],"concepts":[{"id":"https://openalex.org/C2778795013","wikidata":"https://www.wikidata.org/wiki/Q945127","display_name":"Triviality","level":2,"score":0.921796441078186},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8133801817893982},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.7756091356277466},{"id":"https://openalex.org/C146834321","wikidata":"https://www.wikidata.org/wiki/Q2979672","display_name":"Closure (psychology)","level":2,"score":0.6261361837387085},{"id":"https://openalex.org/C47527751","wikidata":"https://www.wikidata.org/wiki/Q1454165","display_name":"Free product","level":3,"score":0.6056808233261108},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.59068763256073},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.5234445929527283},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.5117749571800232},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.44347333908081055},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3201841115951538},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.1034184992313385},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.05841827392578125},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.03694123029708862},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.032551735639572144},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C34447519","wikidata":"https://www.wikidata.org/wiki/Q179522","display_name":"Market economy","level":1,"score":0.0}],"mesh":[],"locations_count":4,"locations":[{"id":"doi:10.1142/s0218196716500223","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218196716500223","pdf_url":null,"source":{"id":"https://openalex.org/S23260266","display_name":"International Journal of Algebra and Computation","issn_l":"0218-1967","issn":["0218-1967","1793-6500"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Algebra and Computation","raw_type":"journal-article"},{"id":"pmh:oai:arXiv.org:1509.02717","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1509.02717","pdf_url":"https://arxiv.org/pdf/1509.02717","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.1509.02717","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1509.02717","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"},{"id":"mag:2267943872","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":null,"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"pmh:oai:arXiv.org:1509.02717","is_oa":true,"landing_page_url":"http://arxiv.org/abs/1509.02717","pdf_url":"https://arxiv.org/pdf/1509.02717","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","score":0.4300000071525574,"display_name":"Reduced inequalities"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320311518","display_name":"Heriot-Watt University","ror":"https://ror.org/04mghma93"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2267943872.pdf","grobid_xml":"https://content.openalex.org/works/W2267943872.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W189240754","https://openalex.org/W1672955954","https://openalex.org/W1974272713","https://openalex.org/W1977775387","https://openalex.org/W1978771091","https://openalex.org/W1982626960","https://openalex.org/W1985126757","https://openalex.org/W1992714457","https://openalex.org/W1994550804","https://openalex.org/W1997104505","https://openalex.org/W1999886840","https://openalex.org/W2011372418","https://openalex.org/W2017671777","https://openalex.org/W2036237111","https://openalex.org/W2048438298","https://openalex.org/W2084183547","https://openalex.org/W2108177195","https://openalex.org/W2122349777","https://openalex.org/W2166847974","https://openalex.org/W2323867948","https://openalex.org/W2990802139","https://openalex.org/W2991108569","https://openalex.org/W4230682442"],"related_works":["https://openalex.org/W2063100212","https://openalex.org/W1989150127","https://openalex.org/W948049815","https://openalex.org/W3103497429","https://openalex.org/W2122349777","https://openalex.org/W2073275713","https://openalex.org/W2126719858","https://openalex.org/W2805181198","https://openalex.org/W2980518627","https://openalex.org/W2031561564","https://openalex.org/W2790391813","https://openalex.org/W2277701423","https://openalex.org/W3146039763","https://openalex.org/W2766996820","https://openalex.org/W2950672018","https://openalex.org/W2038328103","https://openalex.org/W2994648176","https://openalex.org/W2896534785","https://openalex.org/W2335989377","https://openalex.org/W2284337741"],"abstract_inverted_index":{"In":[0,30,49],"this":[1],"paper":[2],"we":[3,32,57],"study":[4],"a":[5,15,27],"group":[6],"[Formula:":[7,44,71],"see":[8,45,72],"text]":[9,46],"which":[10],"is":[11,47],"the":[12,23,36,50,53,59,67],"quotient":[13],"of":[14,18,26,66],"free":[16],"product":[17],"three":[19],"nontrivial":[20],"groups":[21],"by":[22],"normal":[24],"closure":[25],"single":[28],"element.":[29],"particular,":[31],"show":[33],"that":[34,62],"if":[35],"relator":[37],"has":[38],"length":[39],"at":[40,63],"most":[41],"eight,":[42],"then":[43],"nontrivial.":[48],"case":[51],"where":[52],"factors":[54,68],"are":[55],"cyclic,":[56],"prove":[58],"stronger":[60],"result":[61],"least":[64],"one":[65],"embeds":[69],"in":[70],"text].":[73]},"counts_by_year":[],"updated_date":"2026-02-09T09:26:11.010843","created_date":"2025-10-10T00:00:00"}
