{"id":"https://openalex.org/W936422256","doi":"https://doi.org/10.2478/forma-2015-0006","title":"Equivalent Expressions of Direct Sum Decomposition of Groups <sup>1</sup>","display_name":"Equivalent Expressions of Direct Sum Decomposition of Groups <sup>1</sup>","publication_year":2015,"publication_date":"2015-03-01","ids":{"openalex":"https://openalex.org/W936422256","doi":"https://doi.org/10.2478/forma-2015-0006","mag":"936422256"},"language":"en","primary_location":{"id":"doi:10.2478/forma-2015-0006","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2015-0006","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2015-0006","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://sciendo.com/pdf/10.2478/forma-2015-0006","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5005167371","display_name":"Kazuhisa Nakasho","orcid":"https://orcid.org/0000-0003-1110-4342"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Kazuhisa Nakasho","raw_affiliation_strings":["Shinshu University, Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University, Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5048322538","display_name":"Hiroyuki Okazaki","orcid":"https://orcid.org/0009-0000-1022-361X"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Hiroyuki Okazaki","raw_affiliation_strings":["Shinshu University, Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University, Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5015411968","display_name":"Hiroshi Yamazaki","orcid":"https://orcid.org/0000-0001-9434-2957"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Hiroshi Yamazaki","raw_affiliation_strings":["Shinshu University, Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University, Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5055549852","display_name":"Yasunari Shidama","orcid":"https://orcid.org/0000-0001-5634-0838"},"institutions":[{"id":"https://openalex.org/I137975476","display_name":"Shinshu University","ror":"https://ror.org/0244rem06","country_code":"JP","type":"education","lineage":["https://openalex.org/I137975476"]}],"countries":["JP"],"is_corresponding":false,"raw_author_name":"Yasunari Shidama","raw_affiliation_strings":["Shinshu University, Nagano, Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shinshu University, Nagano, Japan","institution_ids":["https://openalex.org/I137975476"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I137975476"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.01260398,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":94},"biblio":{"volume":"23","issue":"1","first_page":"67","last_page":"73"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9995999932289124,"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"}},"topics":[{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.9995999932289124,"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/T11703","display_name":"Rings, Modules, and Algebras","score":0.9952999949455261,"subfield":{"id":"https://openalex.org/subfields/2602","display_name":"Algebra and Number Theory"},"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/T11797","display_name":"graph theory and CDMA systems","score":0.9871000051498413,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.796169638633728},{"id":"https://openalex.org/keywords/decomposition","display_name":"Decomposition","score":0.6545823216438293},{"id":"https://openalex.org/keywords/section","display_name":"Section (typography)","score":0.6486057043075562},{"id":"https://openalex.org/keywords/commutative-property","display_name":"Commutative property","score":0.4670290946960449},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.35812199115753174},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.35132020711898804},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.29358047246932983},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11115455627441406},{"id":"https://openalex.org/keywords/chemistry","display_name":"Chemistry","score":0.0440945029258728}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.796169638633728},{"id":"https://openalex.org/C124681953","wikidata":"https://www.wikidata.org/wiki/Q339062","display_name":"Decomposition","level":2,"score":0.6545823216438293},{"id":"https://openalex.org/C2780129039","wikidata":"https://www.wikidata.org/wiki/Q1931107","display_name":"Section (typography)","level":2,"score":0.6486057043075562},{"id":"https://openalex.org/C183778304","wikidata":"https://www.wikidata.org/wiki/Q165474","display_name":"Commutative property","level":2,"score":0.4670290946960449},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.35812199115753174},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.35132020711898804},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.29358047246932983},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11115455627441406},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0440945029258728},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C111919701","wikidata":"https://www.wikidata.org/wiki/Q9135","display_name":"Operating system","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.2478/forma-2015-0006","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2015-0006","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2015-0006","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:fd0a43c69cfb411b8a085995a58ff850","is_oa":false,"landing_page_url":"https://doaj.org/article/fd0a43c69cfb411b8a085995a58ff850","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Formalized Mathematics, Vol 23, Iss 1, Pp 67-73 (2015)","raw_type":"article"},{"id":"pmh:oai:repozytorium.uwb.edu.pl:11320/4839","is_oa":false,"landing_page_url":"http://hdl.handle.net/11320/4839","pdf_url":null,"source":{"id":"https://openalex.org/S4306400933","display_name":"The Repository of University in Bialystok (University in Bialystok)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210094198","host_organization_name":"University Clinical Hospital In Bialystok","host_organization_lineage":["https://openalex.org/I4210094198"],"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":"Article"}],"best_oa_location":{"id":"doi:10.2478/forma-2015-0006","is_oa":true,"landing_page_url":"https://doi.org/10.2478/forma-2015-0006","pdf_url":"https://sciendo.com/pdf/10.2478/forma-2015-0006","source":{"id":"https://openalex.org/S179241345","display_name":"Formalized Mathematics","issn_l":"1426-2630","issn":["1426-2630","1898-9934"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320322","host_organization_name":"De Gruyter Open","host_organization_lineage":["https://openalex.org/P4310320322","https://openalex.org/P4310313990"],"host_organization_lineage_names":["De Gruyter Open","De Gruyter"],"type":"journal"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Formalized Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","display_name":"Reduced inequalities","score":0.41999998688697815}],"awards":[{"id":"https://openalex.org/G5886570192","display_name":"Content Development for Distance Education in Advanced University Mathematics Using Mizar","funder_award_id":"22300285","funder_id":"https://openalex.org/F4320334764","funder_display_name":"Japan Society for the Promotion of Science"}],"funders":[{"id":"https://openalex.org/F4320334764","display_name":"Japan Society for the Promotion of Science","ror":"https://ror.org/00hhkn466"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W936422256.pdf","grobid_xml":"https://content.openalex.org/works/W936422256.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W22141270","https://openalex.org/W109369165","https://openalex.org/W180239714","https://openalex.org/W192809544","https://openalex.org/W260340778","https://openalex.org/W604140817","https://openalex.org/W1127721947","https://openalex.org/W1541824250","https://openalex.org/W1543877612","https://openalex.org/W1982733235","https://openalex.org/W1986748099","https://openalex.org/W2104238184","https://openalex.org/W2105559527","https://openalex.org/W2118242375","https://openalex.org/W2143507504","https://openalex.org/W2148145049","https://openalex.org/W2150348213","https://openalex.org/W2163090560","https://openalex.org/W2163641194","https://openalex.org/W2296232473","https://openalex.org/W2318276991","https://openalex.org/W2338095303","https://openalex.org/W4230191592","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W2560853036","https://openalex.org/W2566696415","https://openalex.org/W4249026152","https://openalex.org/W1563787543","https://openalex.org/W4241545006","https://openalex.org/W2909241626","https://openalex.org/W2890304493","https://openalex.org/W590015412","https://openalex.org/W4211144708","https://openalex.org/W4211021113"],"abstract_inverted_index":{"Summary":[0],"In":[1,17,42,58],"this":[2],"article,":[3],"the":[4,8,18,23,26,43,59,65,83],"equivalent":[5,49],"expressions":[6],"of":[7,12,39,51,55],"direct":[9,28,53,67,72],"sum":[10,29,54,68],"decomposition":[11,30],"groups":[13],"are":[14],"mainly":[15],"discussed.":[16],"first":[19],"section,":[20,45,61],"we":[21,46,62],"formalize":[22,47,63],"fact":[24],"that":[25,64],"internal":[27,52,71],"can":[31],"be":[32],"defined":[33],"as":[34],"normal":[35],"subgroups":[36],"and":[37,80],"some":[38],"their":[40],"properties.":[41],"second":[44],"an":[48,70],"form":[50],"commutative":[56],"groups.":[57],"last":[60],"external":[66],"leads":[69],"sum.":[73],"We":[74],"referred":[75],"to":[76],"[19],":[77],"[18]":[78],"[8]":[79],"[14]":[81],"in":[82],"formalization.":[84]},"counts_by_year":[{"year":2023,"cited_by_count":1}],"updated_date":"2026-06-26T08:34:08.712188","created_date":"2025-10-10T00:00:00"}
