{"id":"https://openalex.org/W2973028671","doi":"https://doi.org/10.3390/sym11091147","title":"On Finite Quasi-Core-p p-Groups","display_name":"On Finite Quasi-Core-p p-Groups","publication_year":2019,"publication_date":"2019-09-10","ids":{"openalex":"https://openalex.org/W2973028671","doi":"https://doi.org/10.3390/sym11091147","mag":"2973028671"},"language":"en","primary_location":{"id":"doi:10.3390/sym11091147","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11091147","pdf_url":"https://www.mdpi.com/2073-8994/11/9/1147/pdf?version=1573094985","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2073-8994/11/9/1147/pdf?version=1573094985","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5008402094","display_name":"Jiao Wang","orcid":"https://orcid.org/0000-0002-6335-3183"},"institutions":[{"id":"https://openalex.org/I4210102575","display_name":"TianjinSino-German University of Applied Sciences","ror":"https://ror.org/01243ws42","country_code":"CN","type":"education","lineage":["https://openalex.org/I4210102575"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Jiao Wang","raw_affiliation_strings":["Basic Course Department, Tianjin Sino-German University of Applied Sciences, Tianjin 300350, China"],"raw_orcid":"https://orcid.org/0000-0002-6335-3183","affiliations":[{"raw_affiliation_string":"Basic Course Department, Tianjin Sino-German University of Applied Sciences, Tianjin 300350, China","institution_ids":["https://openalex.org/I4210102575"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5040371510","display_name":"Xiuyun Guo","orcid":"https://orcid.org/0000-0003-2288-2983"},"institutions":[{"id":"https://openalex.org/I113940042","display_name":"Shanghai University","ror":"https://ror.org/006teas31","country_code":"CN","type":"education","lineage":["https://openalex.org/I113940042"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xiuyun Guo","raw_affiliation_strings":["Department of Mathematics, Shanghai University, Shanghai 200444, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Shanghai University, Shanghai 200444, China","institution_ids":["https://openalex.org/I113940042"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5008402094"],"corresponding_institution_ids":["https://openalex.org/I4210102575"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":0.506,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.62073365,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"11","issue":"9","first_page":"1147","last_page":"1147"},"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/T11797","display_name":"graph theory and CDMA systems","score":0.9932000041007996,"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"}},{"id":"https://openalex.org/T11130","display_name":"Coding theory and cryptography","score":0.9861999750137329,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7726866602897644},{"id":"https://openalex.org/keywords/commutator-subgroup","display_name":"Commutator subgroup","score":0.6876816153526306},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.6805843114852905},{"id":"https://openalex.org/keywords/prime","display_name":"Prime (order theory)","score":0.5629481077194214},{"id":"https://openalex.org/keywords/core","display_name":"Core (optical fiber)","score":0.5607678294181824},{"id":"https://openalex.org/keywords/normal-subgroup","display_name":"Normal subgroup","score":0.5446344017982483},{"id":"https://openalex.org/keywords/p-group","display_name":"p-group","score":0.540067195892334},{"id":"https://openalex.org/keywords/nilpotent-group","display_name":"Nilpotent group","score":0.524834930896759},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.5069282650947571},{"id":"https://openalex.org/keywords/order","display_name":"Order (exchange)","score":0.5034469962120056},{"id":"https://openalex.org/keywords/exponent","display_name":"Exponent","score":0.5000274181365967},{"id":"https://openalex.org/keywords/abelian-group","display_name":"Abelian group","score":0.4867432117462158},{"id":"https://openalex.org/keywords/finite-group","display_name":"Finite group","score":0.45687806606292725},{"id":"https://openalex.org/keywords/integer","display_name":"Integer (computer science)","score":0.4557390809059143},{"id":"https://openalex.org/keywords/omega-and-agemo-subgroup","display_name":"Omega and agemo subgroup","score":0.41895201802253723},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.3690943121910095},{"id":"https://openalex.org/keywords/torsion-subgroup","display_name":"Torsion subgroup","score":0.3386036157608032},{"id":"https://openalex.org/keywords/nilpotent","display_name":"Nilpotent","score":0.26435911655426025},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.24365490674972534},{"id":"https://openalex.org/keywords/elementary-abelian-group","display_name":"Elementary abelian group","score":0.12637490034103394}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7726866602897644},{"id":"https://openalex.org/C140865510","wikidata":"https://www.wikidata.org/wiki/Q522216","display_name":"Commutator subgroup","level":4,"score":0.6876816153526306},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6805843114852905},{"id":"https://openalex.org/C184992742","wikidata":"https://www.wikidata.org/wiki/Q7243229","display_name":"Prime (order theory)","level":2,"score":0.5629481077194214},{"id":"https://openalex.org/C2164484","wikidata":"https://www.wikidata.org/wiki/Q5170150","display_name":"Core (optical fiber)","level":2,"score":0.5607678294181824},{"id":"https://openalex.org/C105515060","wikidata":"https://www.wikidata.org/wiki/Q743179","display_name":"Normal subgroup","level":3,"score":0.5446344017982483},{"id":"https://openalex.org/C160826032","wikidata":"https://www.wikidata.org/wiki/Q286972","display_name":"p-group","level":3,"score":0.540067195892334},{"id":"https://openalex.org/C206219070","wikidata":"https://www.wikidata.org/wiki/Q1755242","display_name":"Nilpotent group","level":3,"score":0.524834930896759},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.5069282650947571},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5034469962120056},{"id":"https://openalex.org/C2780388253","wikidata":"https://www.wikidata.org/wiki/Q5421508","display_name":"Exponent","level":2,"score":0.5000274181365967},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.4867432117462158},{"id":"https://openalex.org/C2777404646","wikidata":"https://www.wikidata.org/wiki/Q1057968","display_name":"Finite group","level":3,"score":0.45687806606292725},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4557390809059143},{"id":"https://openalex.org/C146531340","wikidata":"https://www.wikidata.org/wiki/Q7090038","display_name":"Omega and agemo subgroup","level":5,"score":0.41895201802253723},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3690943121910095},{"id":"https://openalex.org/C14669601","wikidata":"https://www.wikidata.org/wiki/Q2293759","display_name":"Torsion subgroup","level":4,"score":0.3386036157608032},{"id":"https://openalex.org/C50555996","wikidata":"https://www.wikidata.org/wiki/Q840023","display_name":"Nilpotent","level":2,"score":0.26435911655426025},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.24365490674972534},{"id":"https://openalex.org/C81008192","wikidata":"https://www.wikidata.org/wiki/Q1017231","display_name":"Elementary abelian group","level":3,"score":0.12637490034103394},{"id":"https://openalex.org/C10138342","wikidata":"https://www.wikidata.org/wiki/Q43015","display_name":"Finance","level":1,"score":0.0},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/sym11091147","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11091147","pdf_url":"https://www.mdpi.com/2073-8994/11/9/1147/pdf?version=1573094985","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:80d9350c627e491d8e9bc6ee02044f46","is_oa":true,"landing_page_url":"https://doaj.org/article/80d9350c627e491d8e9bc6ee02044f46","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":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Symmetry, Vol 11, Iss 9, p 1147 (2019)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/sym11091147","is_oa":true,"landing_page_url":"https://doi.org/10.3390/sym11091147","pdf_url":"https://www.mdpi.com/2073-8994/11/9/1147/pdf?version=1573094985","source":{"id":"https://openalex.org/S190787756","display_name":"Symmetry","issn_l":"2073-8994","issn":["2073-8994"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Symmetry","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2973028671.pdf","grobid_xml":"https://content.openalex.org/works/W2973028671.grobid-xml"},"referenced_works_count":10,"referenced_works":["https://openalex.org/W1990790861","https://openalex.org/W2013442873","https://openalex.org/W2047364045","https://openalex.org/W2079522203","https://openalex.org/W2104069672","https://openalex.org/W2163892986","https://openalex.org/W2490728539","https://openalex.org/W2587935764","https://openalex.org/W3150780854","https://openalex.org/W6733111879"],"related_works":["https://openalex.org/W1995094252","https://openalex.org/W2888019080","https://openalex.org/W1975687020","https://openalex.org/W2359877275","https://openalex.org/W4386552579","https://openalex.org/W1567576179","https://openalex.org/W2067205437","https://openalex.org/W3206411551","https://openalex.org/W1997960464","https://openalex.org/W2266963015"],"abstract_inverted_index":{"Given":[0],"a":[1,5,66,133],"positive":[2],"integer":[3],"n,":[4],"finite":[6,55],"group":[7],"G":[8,20,36],"is":[9,37,69,90,96,119,136],"called":[10],"quasi-core-n":[11],"if":[12,61],"\u27e8":[13,17,33,42],"x":[14,18,29,34,43],"\u27e9":[15,19,35,44],"/":[16],"has":[21,109],"order":[22],"at":[23,112,122],"most":[24,113,123],"n":[25],"for":[26],"any":[27],"element":[28],"in":[30,45],"G,":[31],"where":[32,88],"the":[38,52,62,75,129],"normal":[39],"core":[40],"of":[41,54,65,77,120,132],"G.":[46],"In":[47],"this":[48],"paper,":[49],"we":[50,103],"investigate":[51],"structure":[53],"quasi-core-p":[56,67],"p-groups.":[57],"We":[58,125],"prove":[59,104],"that":[60,105,128],"nilpotency":[63,110],"class":[64,111],"p-group":[68],"p":[70,83,89,99],"+":[71,85],"m":[72,84,95],",":[73,87,102],"then":[74],"exponent":[76,121],"its":[78,116],"commutator":[79,117],"subgroup":[80,118,131],"cannot":[81],"exceed":[82],"1":[86],"an":[91],"odd":[92],"prime":[93],"and":[94,115],"non-negative.":[97],"If":[98],"=":[100],"3":[101],"every":[106],"quasi-core-3":[107],"3-group":[108],"5":[114],"9.":[124],"also":[126],"show":[127],"Frattini":[130],"quasi-core-2":[134],"2-group":[135],"abelian.":[137]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2019-09-19T00:00:00"}
