{"id":"https://openalex.org/W7212030264","doi":"https://doi.org/10.48550/arxiv.2609.05805","title":"Odd square roots and the sum of element orders on Sn and An","display_name":"Odd square roots and the sum of element orders on Sn and An","publication_year":2026,"publication_date":"2026-09-05","ids":{"openalex":"https://openalex.org/W7212030264","doi":"https://doi.org/10.48550/arxiv.2609.05805"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.05805","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.05805","pdf_url":"https://arxiv.org/pdf/2609.05805","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/2609.05805","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101680285","display_name":"Manoj Kumar Singh","orcid":"https://orcid.org/0000-0002-7687-3970","observed_orcids":["https://orcid.org/0000-0002-7687-3970"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Singh, Manoj Kumar","raw_affiliation_strings":["Department of Mathematics and Statistics , St. Xavier's College , Ranchi , Jharkhand 834001 , India"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Statistics , St. Xavier's College , Ranchi , Jharkhand 834001 , India","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5101680285"],"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":0.9951000213623047,"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":0.9951000213623047,"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":0.004600000102072954,"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/T12504","display_name":"Mathematics and Applications","score":9.999999747378752e-05,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/symmetric-group","display_name":"symmetric group","score":0.7960000038146973},{"id":"https://openalex.org/keywords/alternating-groups","display_name":"alternating groups","score":0.9990000128746033},{"id":"https://openalex.org/keywords/finite-groups","display_name":"finite groups","score":0.39800000190734863}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8683000206947327},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7261999845504761},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.6692000031471252},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.5849999785423279},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.548799991607666},{"id":"https://openalex.org/C199422724","wikidata":"https://www.wikidata.org/wiki/Q41118","display_name":"Quotient","level":2,"score":0.499099999666214},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.4693000018596649},{"id":"https://openalex.org/C2777151079","wikidata":"https://www.wikidata.org/wiki/Q141160","display_name":"Parity (physics)","level":2,"score":0.46160000562667847},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.46070000529289246},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4546999931335449},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.37860000133514404},{"id":"https://openalex.org/C184264201","wikidata":"https://www.wikidata.org/wiki/Q21199","display_name":"Natural number","level":2,"score":0.3749000132083893},{"id":"https://openalex.org/C11577676","wikidata":"https://www.wikidata.org/wiki/Q134237","display_name":"Square root","level":2,"score":0.2980000078678131},{"id":"https://openalex.org/C2778158742","wikidata":"https://www.wikidata.org/wiki/Q1665115","display_name":"Rayleigh quotient","level":3,"score":0.2567000091075897},{"id":"https://openalex.org/C182964748","wikidata":"https://www.wikidata.org/wiki/Q208216","display_name":"Triangle inequality","level":2,"score":0.2524000108242035},{"id":"https://openalex.org/C92957085","wikidata":"https://www.wikidata.org/wiki/Q1412905","display_name":"Permutation group","level":3,"score":0.25200000405311584},{"id":"https://openalex.org/C64137656","wikidata":"https://www.wikidata.org/wiki/Q1064405","display_name":"Parity of a permutation","level":4,"score":0.25029999017715454}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.05805","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.05805","pdf_url":"https://arxiv.org/pdf/2609.05805","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.2609.05805","doi":"https://doi.org/10.48550/arxiv.2609.05805","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.05805","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-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.05805","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.05805","pdf_url":"https://arxiv.org/pdf/2609.05805","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":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7212030264.pdf","grobid_xml":"https://content.openalex.org/works/W7212030264.grobid-xml"},"referenced_works_count":4,"referenced_works":["https://openalex.org/W2329436721","https://openalex.org/W4401524282","https://openalex.org/W4416190669","https://openalex.org/W7202273621"],"related_works":["https://openalex.org/W2068528259","https://openalex.org/W2345504666","https://openalex.org/W2032983827","https://openalex.org/W2370645957","https://openalex.org/W4312607299","https://openalex.org/W2184501533","https://openalex.org/W4233477301","https://openalex.org/W2156190203","https://openalex.org/W4394781015","https://openalex.org/W4390021869"],"abstract_inverted_index":{"For":[0],"a":[1,46],"finite":[2],"group":[3],"$G$":[4],"let":[5],"$\u03c8(G)$":[6],"be":[7,27,56],"the":[8,11,49,57,68,71,74,95,107,118,135,142],"sum":[9,69],"of":[10,13,51,59,70,73,110,121],"orders":[12,72],"its":[14],"elements.":[15],"The":[16,24,88,147],"quotient":[17],"$S_n/A_n$":[18],"has":[19],"order":[20,96,109,120,143],"two.":[21],"Hence":[22],"$\u03c8(S_n/A_n)=3$.":[23],"quantity":[25],"to":[26,45,164],"compared":[28],"with":[29,63],"$\u03c8(S_n)$":[30],"is":[31,114],"therefore":[32],"$3\u03c8(A_n)$.":[33],"Computation":[34],"indicates":[35],"that":[36,67,83,92,106],"$3\u03c8(A_n)>\u03c8(S_n)$":[37],"for":[38],"every":[39,145],"$n\\ge3$.":[40],"We":[41,65,80,125],"reduce":[42],"this":[43],"inequality":[44,89],"question":[47],"on":[48,101],"parity":[50],"square":[52],"roots.":[53],"Let":[54],"$r(\u03b2)$":[55],"number":[58],"odd":[60,75,112,136],"permutations":[61,76,137],"$\u03c3$":[62],"$\u03c3^{2}=\u03b2$.":[64],"show":[66,82],"equals":[77,86],"$2\\sum_{\u03b2\\in":[78],"A_n}r(\u03b2)o(\u03b2)$.":[79],"also":[81],"$\\sum_{\u03b2\\in":[84],"A_n}r(\u03b2)$":[85],"$|A_n|$.":[87],"then":[90,126],"says":[91,104],"$r$":[93],"and":[94],"function":[97,158],"are":[98,166],"negatively":[99],"correlated":[100],"$A_n$.":[102],"It":[103],"equally":[105],"average":[108,119],"an":[111,122],"permutation":[113],"less":[115],"than":[116],"twice":[117],"even":[123],"one.":[124],"rule":[127],"out":[128],"two":[129],"natural":[130],"approaches.":[131],"No":[132],"injection":[133],"from":[134],"into":[138],"$A_n$":[139],"can":[140],"halve":[141],"at":[144,151],"point.":[146],"smallest":[148],"failure":[149],"occurs":[150],"$n=12$.":[152],"A":[153],"threshold":[154],"argument":[155],"through":[156],"Landau's":[157],"fails":[159],"as":[160],"well.":[161],"Computations":[162],"up":[163],"$n=60$":[165],"reported.":[167]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-10T00:00:00"}
