{"id":"https://openalex.org/W7221542030","doi":"https://doi.org/10.48550/arxiv.2610.12912","title":"Product and Quotient Set Dominance in Finite Groups","display_name":"Product and Quotient Set Dominance in Finite Groups","publication_year":2026,"publication_date":"2026-10-09","ids":{"openalex":"https://openalex.org/W7221542030","doi":"https://doi.org/10.48550/arxiv.2610.12912"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.12912","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.12912","pdf_url":"https://arxiv.org/pdf/2610.12912","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.12912","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5145988461","display_name":"Yuval Amit","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Amit, Yuval","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5035260139","display_name":"Lucas Chen","orcid":"https://orcid.org/0009-0001-3480-4264","observed_orcids":["https://orcid.org/0009-0001-3480-4264"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chen, Lucas","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5114369593","display_name":"Christopher Housholder","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Housholder, Christopher","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5147326134","display_name":"Joshua Im","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Im, Joshua","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5089003707","display_name":"Joshua Khan","orcid":"https://orcid.org/0000-0002-3391-9457","observed_orcids":["https://orcid.org/0000-0002-3391-9457"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Khan, Joshua","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5109586747","display_name":"Steven J. Miller","orcid":"https://orcid.org/0000-0003-4904-4605","observed_orcids":["https://orcid.org/0000-0003-4904-4605"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Miller, Steven J.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5158494069","display_name":"Mateo Palomares","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Palomares, Mateo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5147324366","display_name":"Devayani Pradhan","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pradhan, Devayani","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.86080002784729,"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":0.86080002784729,"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.093299999833107,"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/T10948","display_name":"Advanced Combinatorial Mathematics","score":0.026100000366568565,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/product-sets","display_name":"product sets","score":0.5659999847412109},{"id":"https://openalex.org/keywords/quotient-sets","display_name":"quotient sets","score":0.8379999995231628},{"id":"https://openalex.org/keywords/finite-groups","display_name":"finite groups","score":0.6589999794960022},{"id":"https://openalex.org/keywords/dihedral-groups","display_name":"dihedral groups","score":0.9819999933242798},{"id":"https://openalex.org/keywords/symmetric-group","display_name":"symmetric group","score":0.7710000276565552},{"id":"https://openalex.org/keywords/abelian-groups","display_name":"abelian groups","score":0.9330000281333923},{"id":"https://openalex.org/keywords/independent-sets","display_name":"independent sets","score":0.5559999942779541},{"id":"https://openalex.org/keywords/moments-of-random-variables","display_name":"moments of random variables","score":0.6579999923706055},{"id":"https://openalex.org/keywords/irreducible-characters","display_name":"irreducible characters","score":0.796999990940094},{"id":"https://openalex.org/keywords/conjugacy-classes","display_name":"conjugacy classes","score":0.871999979019165},{"id":"https://openalex.org/keywords/central-elements","display_name":"central elements","score":0.414000004529953}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8995000123977661},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.722599983215332},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5842000246047974},{"id":"https://openalex.org/C87945829","wikidata":"https://www.wikidata.org/wiki/Q1353233","display_name":"Conjugacy class","level":2,"score":0.5464000105857849},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.5378000140190125},{"id":"https://openalex.org/C136170076","wikidata":"https://www.wikidata.org/wiki/Q181296","display_name":"Abelian group","level":2,"score":0.5223000049591064},{"id":"https://openalex.org/C2777404646","wikidata":"https://www.wikidata.org/wiki/Q1057968","display_name":"Finite group","level":3,"score":0.5088000297546387},{"id":"https://openalex.org/C128622974","wikidata":"https://www.wikidata.org/wiki/Q849512","display_name":"Symmetric group","level":2,"score":0.4575999975204468},{"id":"https://openalex.org/C90673727","wikidata":"https://www.wikidata.org/wiki/Q901718","display_name":"Product (mathematics)","level":2,"score":0.4156000018119812},{"id":"https://openalex.org/C63388996","wikidata":"https://www.wikidata.org/wiki/Q1340623","display_name":"Classification of finite simple groups","level":4,"score":0.4058000147342682},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.39489999413490295},{"id":"https://openalex.org/C72409365","wikidata":"https://www.wikidata.org/wiki/Q558339","display_name":"Dihedral group","level":3,"score":0.3912999927997589},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.3878999948501587},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.30809998512268066},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.29910001158714294},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.273499995470047},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.27320000529289246},{"id":"https://openalex.org/C51997251","wikidata":"https://www.wikidata.org/wiki/Q1406425","display_name":"Group of Lie type","level":3,"score":0.2685999870300293},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.2671999931335449},{"id":"https://openalex.org/C20725272","wikidata":"https://www.wikidata.org/wiki/Q245462","display_name":"Cyclic group","level":3,"score":0.26589998602867126},{"id":"https://openalex.org/C95627345","wikidata":"https://www.wikidata.org/wiki/Q1138961","display_name":"Quotient group","level":4,"score":0.26109999418258667},{"id":"https://openalex.org/C18116961","wikidata":"https://www.wikidata.org/wiki/Q734209","display_name":"Generating set of a group","level":2,"score":0.25760000944137573},{"id":"https://openalex.org/C58193220","wikidata":"https://www.wikidata.org/wiki/Q1778193","display_name":"Direct product","level":2,"score":0.2565999925136566},{"id":"https://openalex.org/C2777044963","wikidata":"https://www.wikidata.org/wiki/Q5384707","display_name":"Equivalence class (music)","level":2,"score":0.25060001015663147}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.12912","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.12912","pdf_url":"https://arxiv.org/pdf/2610.12912","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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.2610.12912","doi":"https://doi.org/10.48550/arxiv.2610.12912","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.12912","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.12912","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.12912","pdf_url":"https://arxiv.org/pdf/2610.12912","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G7041586996","display_name":"Conference: The Polymath Jr Program","funder_award_id":"2341670","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320308845","display_name":"Williams College","ror":"https://ror.org/04avkmd49"},{"id":"https://openalex.org/F4320310246","display_name":"University of Rochester","ror":"https://ror.org/022kthw22"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7221542030.pdf","grobid_xml":"https://content.openalex.org/works/W7221542030.grobid-xml"},"referenced_works_count":15,"referenced_works":["https://openalex.org/W1973882569","https://openalex.org/W2000713788","https://openalex.org/W2028491355","https://openalex.org/W2045099620","https://openalex.org/W2048305092","https://openalex.org/W2097610146","https://openalex.org/W2111033825","https://openalex.org/W2138081888","https://openalex.org/W2164181738","https://openalex.org/W2802971187","https://openalex.org/W2951614950","https://openalex.org/W4213167573","https://openalex.org/W6950057416","https://openalex.org/W7128614670","https://openalex.org/W7204185638"],"related_works":["https://openalex.org/W2167815714","https://openalex.org/W4210569233","https://openalex.org/W4414759628","https://openalex.org/W2031452634","https://openalex.org/W2497091326","https://openalex.org/W2320124569","https://openalex.org/W2028668049","https://openalex.org/W1598295920","https://openalex.org/W2488434289","https://openalex.org/W3037715314"],"abstract_inverted_index":{"A":[0,47],"classic":[1],"problem":[2],"in":[3],"additive":[4],"number":[5,158],"theory":[6],"asks":[7],"whether":[8],"a":[9,42,56,121],"given":[10],"subset":[11,43],"of":[12,31,59,81,89,124,159,198,202,217,251],"integers":[13],"has":[14],"more":[15],"sums":[16],"or":[17],"differences.":[18],"This":[19],"inquiry":[20],"naturally":[21],"extends":[22],"to":[23,76],"finite":[24,48,83,125,210],"groups,":[25,203],"where":[26,61],"one":[27],"compares":[28],"the":[29,32,37,68,78,87,90,136,142,157,165,176,195,241,245,252],"sizes":[30],"product":[33,100],"set":[34,39],"$AA$":[35],"and":[36,65,101,132,208,220,248],"quotient":[38,102],"$AA^{-1}$":[40],"for":[41,120,186,227,240],"$A":[44],"\\subseteq":[45],"G$.":[46],"group":[49],"is":[50,70,163],"considered":[51],"product-dominant":[52],"if":[53,67,148],"it":[54],"contains":[55],"higher":[57],"proportion":[58],"subsets":[60],"$|AA|":[62],">":[63],"|AA^{-1}|$,":[64],"quotient-dominant":[66],"reverse":[69],"true.":[71],"We":[72],"establish":[73],"statistical":[74],"criteria":[75],"determine":[77],"dominance":[79,196],"behavior":[80,197],"large":[82],"groups":[84,126,143,166,184,207,212,216,223,228],"by":[85,179],"analyzing":[86],"moments":[88,110],"random":[91],"variable":[92],"$d(A)":[93],":=":[94],"|AA|":[95],"-":[96],"|AA^{-1}|$.":[97],"By":[98],"translating":[99],"relationships":[103],"into":[104],"graph-theoretic":[105],"constructions,":[106],"we":[107,174,193,235],"compute":[108],"these":[109,171],"through":[111],"counting":[112],"independent":[113],"sets.":[114],"Our":[115],"main":[116],"result":[117],"demonstrates":[118],"that":[119,181,205,229],"growing":[122],"sequence":[123],"$G_k$":[127],"with":[128],"center":[129],"size":[130],"$Z_k$":[131],"involution":[133],"count":[134],"$T_k$,":[135],"condition":[137],"$Z_k":[138],"=":[139,152],"o(T_k)$":[140],"implies":[141],"are":[144,167,190,213,224],"eventually":[145,168],"product-dominant.":[146,191],"Conversely,":[147],"$T_k":[149],"\\cdot":[150],"\\sqrt{3}^{R_k/Z_k}":[151],"o(Z_k)$":[153],"(where":[154],"$R_k$":[155],"denotes":[156],"elements":[160],"whose":[161],"square":[162],"central),":[164],"quotient-dominant.":[169,225],"Using":[170],"asymptotic":[172,233],"conditions,":[173,234],"resolve":[175],"Miller--Vissuet":[177],"conjecture":[178],"proving":[180,204],"all":[182],"dihedral":[183],"$D_n$":[185],"$n":[187],"\\geq":[188],"3$":[189],"Furthermore,":[192],"classify":[194],"several":[199],"major":[200],"families":[201],"symmetric":[206],"non-abelian":[209],"simple":[211],"product-dominant,":[214],"while":[215],"odd":[218],"order":[219],"certain":[221],"abelian":[222],"Finally,":[226],"fall":[230],"outside":[231],"our":[232],"derive":[236],"an":[237],"explicit":[238],"formula":[239],"first":[242],"moment":[243],"utilizing":[244],"irreducible":[246],"characters":[247],"conjugacy":[249],"classes":[250],"group.":[253]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-13T00:00:00"}