{"id":"https://openalex.org/W2144464157","doi":"https://doi.org/10.37236/712","title":"Invariant Relations and Aschbacher Classes of Finite Linear Groups","display_name":"Invariant Relations and Aschbacher Classes of Finite Linear Groups","publication_year":2011,"publication_date":"2011-11-21","ids":{"openalex":"https://openalex.org/W2144464157","doi":"https://doi.org/10.37236/712","mag":"2144464157"},"language":"en","primary_location":{"id":"doi:10.37236/712","is_oa":true,"landing_page_url":"https://doi.org/10.37236/712","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p225/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p225/pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5013844832","display_name":"Jing Xu","orcid":"https://orcid.org/0000-0003-0388-5046"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Jing Xu","raw_affiliation_strings":["School of Mathematics and Statistics"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5071017819","display_name":"Michael Giudici","orcid":"https://orcid.org/0000-0001-5412-4656"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Michael Giudici","raw_affiliation_strings":["School of Mathematics and Statistics"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5074348636","display_name":"Cai Heng Li","orcid":"https://orcid.org/0000-0001-8917-9323"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cai Heng Li","raw_affiliation_strings":["School of Mathematics and Statistics"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5043525348","display_name":"Cheryl E. Praeger","orcid":"https://orcid.org/0000-0002-0881-7336"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cheryl E. Praeger","raw_affiliation_strings":["School of Mathematics and Statistics"],"affiliations":[{"raw_affiliation_string":"School of Mathematics and Statistics","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5013844832"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":true,"cited_by_count":13,"citation_normalized_percentile":{"value":0.19666048,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"18","issue":"1","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":0.9998999834060669,"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":0.9998999834060669,"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":0.9969000220298767,"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/T11130","display_name":"Coding theory and cryptography","score":0.9926999807357788,"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/combinatorics","display_name":"Combinatorics","score":0.7779418230056763},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6980510950088501},{"id":"https://openalex.org/keywords/omega","display_name":"Omega","score":0.6050618290901184},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.6014758348464966},{"id":"https://openalex.org/keywords/cartesian-product","display_name":"Cartesian product","score":0.45953866839408875},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.33211493492126465},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.27033406496047974},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.09858366847038269}],"concepts":[{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7779418230056763},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6980510950088501},{"id":"https://openalex.org/C2779557605","wikidata":"https://www.wikidata.org/wiki/Q9890","display_name":"Omega","level":2,"score":0.6050618290901184},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.6014758348464966},{"id":"https://openalex.org/C65236422","wikidata":"https://www.wikidata.org/wiki/Q173740","display_name":"Cartesian product","level":2,"score":0.45953866839408875},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.33211493492126465},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.27033406496047974},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.09858366847038269},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.37236/712","is_oa":true,"landing_page_url":"https://doi.org/10.37236/712","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p225/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},{"id":"pmh:oai:pure.atira.dk:publications/49f9209c-867d-4539-86bf-7e2ce3dba35a","is_oa":false,"landing_page_url":"https://research-repository.uwa.edu.au/en/publications/49f9209c-867d-4539-86bf-7e2ce3dba35a","pdf_url":null,"source":{"id":"https://openalex.org/S4306402492","display_name":"UWA Profiles and Research Repository (UWA)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I177877127","host_organization_name":"The University of Western Australia","host_organization_lineage":["https://openalex.org/I177877127"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Xu, J, Li, C-H, Giudici, M & Praeger, C 2011, 'Invariant relations and Aschbacher classes of finite linear groups', The Electronic Journal of Combinatorics, vol. 18, pp. 1-33.","raw_type":"info:eu-repo/semantics/publishedVersion"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.372.7053","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.372.7053","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.combinatorics.org/ojs/index.php/eljc/article/viewFile/v18i1p225/pdf/","raw_type":"text"}],"best_oa_location":{"id":"doi:10.37236/712","is_oa":true,"landing_page_url":"https://doi.org/10.37236/712","pdf_url":"https://www.combinatorics.org/ojs/index.php/eljc/article/download/v18i1p225/pdf","source":{"id":"https://openalex.org/S38448739","display_name":"The Electronic Journal of Combinatorics","issn_l":"1077-8926","issn":["1077-8926","1097-1440"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310317970","host_organization_name":"Electronic Journal of Combinatorics","host_organization_lineage":["https://openalex.org/P4310317970"],"host_organization_lineage_names":["Electronic Journal of Combinatorics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"The Electronic Journal of Combinatorics","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2144464157.pdf","grobid_xml":"https://content.openalex.org/works/W2144464157.grobid-xml"},"referenced_works_count":24,"referenced_works":["https://openalex.org/W12766527","https://openalex.org/W604130584","https://openalex.org/W1493654001","https://openalex.org/W1506076469","https://openalex.org/W1530075677","https://openalex.org/W1964582653","https://openalex.org/W1968363362","https://openalex.org/W1988155197","https://openalex.org/W1989418639","https://openalex.org/W1991451613","https://openalex.org/W2008730668","https://openalex.org/W2014063324","https://openalex.org/W2062649746","https://openalex.org/W2089950455","https://openalex.org/W2093674973","https://openalex.org/W2102026239","https://openalex.org/W2114764790","https://openalex.org/W2165263868","https://openalex.org/W2318618198","https://openalex.org/W2490728539","https://openalex.org/W2502697445","https://openalex.org/W2798943694","https://openalex.org/W4214866377","https://openalex.org/W4231126827"],"related_works":["https://openalex.org/W2999042429","https://openalex.org/W2740105782","https://openalex.org/W2080220036","https://openalex.org/W2090448654","https://openalex.org/W2016962652","https://openalex.org/W2741635787","https://openalex.org/W2766376282","https://openalex.org/W2191376159","https://openalex.org/W2952539738","https://openalex.org/W1006332129"],"abstract_inverted_index":{"For":[0],"a":[1,5,8,12,25,28,116,181,217],"positive":[2],"integer":[3],"$k$,":[4],"$k$-relation":[6,26,184],"on":[7,32,56,88],"set":[9,61,70,90],"$\\Omega$":[10,33,57,91],"is":[11,23,50,66,156],"non-empty":[13],"subset":[14,117],"$\\Delta$":[15,22,37],"of":[16,43,48,62,71,92,95,108,111,121,172,203,220,228],"the":[17,46,51,60,69,89,96,170,223],"$k$-fold":[18],"Cartesian":[19],"product":[20],"$\\Omega^k$;":[21],"called":[24],"for":[27,64,73,78,102],"permutation":[29,53,231],"group":[30,54],"$H$":[31,35,131,140,155,173,204],"if":[34,136,139,154,174,177],"leaves":[36,141,179,194],"invariant":[38,142,180,195],"setwise.":[39],"The":[40],"$k$-closure":[41,171],"$H^{(k)}$":[42],"$H$,":[44],"in":[45,84,133,147,160,169,185,209],"sense":[47],"Wielandt,":[49],"largest":[52],"$K$":[55,65],"such":[58],"that":[59,130,192],"$k$-relations":[63,72,77,122],"equal":[67],"to":[68],"$H$.":[74],"We":[75],"study":[76],"finite":[79],"semi-linear":[80],"groups":[81],"$H\\leq{\\rm\\Gamma":[82],"L}(d,q)$":[83,167,207],"their":[85],"natural":[86],"action":[87],"non-zero":[93],"vectors":[94],"underlying":[97],"vector":[98],"space.":[99],"In":[100],"particular,":[101],"each":[103],"Aschbacher":[104],"class":[105],"${\\mathcal":[106,134,161,210],"C}$":[107,135],"geometric":[109],"subgroups":[110,159],"${\\rm\\Gamma":[112],"L}(d,q)$,":[113],"we":[114,215],"define":[115],"${\\rm":[118,148,186],"Rel}({\\mathcal":[119,149,187],"C})$":[120,188],"(with":[123],"$k=1$":[124],"or":[125,201],"$k=2$)":[126],"and":[127,137,151,175,205,222],"prove":[128],"(i)":[129],"lies":[132,168],"only":[138,176],"at":[143],"least":[144],"one":[145],"relation":[146],"C})$,":[150],"(ii)":[152],"that,":[153],"maximal":[157],"among":[158],"C}$,":[162],"then":[163],"an":[164,213],"element":[165],"$g\\in{\\rm\\Gamma":[166],"$g$":[178,193],"single":[182],"$H$-invariant":[183,197],"(rather":[189],"than":[190],"checking":[191],"all":[196],"$k$-relations).":[198],"Consequently":[199],"both,":[200],"neither,":[202],"$H^{(k)}\\cap{\\rm\\Gamma":[206],"lie":[208],"C}$.":[211],"As":[212],"application,":[214],"improve":[216],"1992":[218],"result":[219],"Saxl":[221],"fourth":[224],"author":[225],"concerning":[226],"closures":[227],"affine":[229],"primitive":[230],"groups.":[232]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":5},{"year":2020,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
