[ { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ne4cd-2a765", "eprint_id": 122406, "eprint_status": "archive", "datestamp": "2023-08-22 17:41:39", "lastmod": "2023-10-20 20:31:45", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" }, "orcid": "0000-0002-8380-0921" } ] }, "title": "Fusion systems with 2-small components", "ispublished": "pub", "full_text_status": "public", "keywords": "Applied Mathematics; General Mathematics", "note": "\u00a9 2023 American Mathematical Society. \n\nThis work was partially supported by DMS NSF-1265587 and DMS NSF-1601063.", "abstract": "We show that there are no odd simple 2-fusion systems F in which the centralizer of some fully centralized involution contains a component C that is the 2-fusion system of a simple group K such that C is J-maximal or maximal and subintrinsic in C(F), as appropriate, and such that K is of Lie type over the field of order 2, but not Sp\u2099(2) or F\u2084(2); or K is one of many sporadic groups; or K is P\u03a9\u207a\u2088(3).", "date": "2023-08-17", "date_type": "published", "publication": "Transactions of the American Mathematical Society", "publisher": "American Mathematical Society", "id_number": "CaltechAUTHORS:20230725-746861000.32", "issn": "0002-9947", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230725-746861000.32", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1265587" }, { "agency": "NSF", "grant_number": "DMS-1601063" } ] }, "doi": "10.1090/tran/8797", "resource_type": "article", "pub_year": "2023", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/vtna3-xc530", "eprint_id": 109463, "eprint_status": "archive", "datestamp": "2023-08-22 17:52:58", "lastmod": "2023-10-23 17:58:58", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" }, "orcid": "0000-0002-8380-0921" } ] }, "title": "Fusion systems with U\u2083(3) J-components", "ispublished": "pub", "full_text_status": "restricted", "keywords": "Fusion systems; Finite simple groups", "note": "\u00a9 2021 Elsevier Inc. \n\nReceived 27 September 2020, Available online 9 June 2021. \n\nThis work was partially supported by DMS NSF-1265587 and DMS NSF-1601063.", "abstract": "We determine the 2-fusion systems of J-component type in which the centralizer of some fully centralized involution has a maximal J-component that is the 2-fusion system of U\u2083(3).", "date": "2022-10-01", "date_type": "published", "publication": "Journal of Algebra", "volume": "607", "publisher": "Elsevier", "pagerange": "34-63", "id_number": "CaltechAUTHORS:20210610-093254072", "issn": "0021-8693", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210610-093254072", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1265587" }, { "agency": "NSF", "grant_number": "DMS-1601063" } ] }, "doi": "10.1016/j.jalgebra.2021.06.006", "resource_type": "article", "pub_year": "2022", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/k0xg6-g5870", "eprint_id": 116703, "eprint_status": "archive", "datestamp": "2023-08-20 06:22:59", "lastmod": "2023-10-24 21:09:18", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" }, "orcid": "0000-0002-8380-0921" } ] }, "title": "Fusion systems with J-components over F-_(2^e) with e > 1", "ispublished": "pub", "full_text_status": "public", "note": "Funding source: National Science Foundation\n\nAward Identifier / Grant number: DMS NSF-1265587\n\nAward Identifier / Grant number: DMS NSF-1601063\n\nFunding statement: This work was partially supported by DMS NSF-1265587 and DMS NSF-1601063.\n\nCommunicated by: Christopher W. Parker", "abstract": "Let \u03ba be a finite simple group of Lie type over a field of even order q > 2. If \u03ba is not \u00b2F\u2084(q), then we determine the fusion systems \u2131 of J-component type with a fully centralized involution j such that C_(\u2131)(j) has a component realized by \u03ba. The exceptional case is treated in a later paper.", "date": "2022", "date_type": "published", "publication": "Journal of Group Theory", "publisher": "De Gruyter", "id_number": "CaltechAUTHORS:20220901-221643366", "issn": "1433-5883", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220901-221643366", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.1515/jgth-2020-0156", "resource_type": "article", "pub_year": "2022", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/fbad7-znn76", "eprint_id": 106264, "eprint_status": "archive", "datestamp": "2023-08-20 02:28:31", "lastmod": "2023-10-20 23:17:30", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" }, "orcid": "0000-0002-8380-0921" } ] }, "title": "Walter's theorem for fusion systems", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 2020 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.\n\nReceived 1 April 2019; revised 25 June 2020; published online 23 October 2020.\n\nThis work was partially supported by DMS NSF-1265587 and DMS NSF-1601063.", "abstract": "We determine the saturated 2\u2010fusion systems in which the centralizer of some fully centralized involution contains a component that is the 2\u2010fusion system of a large group of Lie type over a field of odd order.", "date": "2021-04", "date_type": "published", "publication": "Proceedings of the London Mathematical Society", "volume": "122", "number": "4", "publisher": "London Mathematical Society", "pagerange": "569-615", "id_number": "CaltechAUTHORS:20201023-130934536", "issn": "0024-6115", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20201023-130934536", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1265587" }, { "agency": "NSF", "grant_number": "DMS-1601063" } ] }, "collection": "CaltechAUTHORS", "doi": "10.1112/plms.12386", "resource_type": "article", "pub_year": "2021", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/2ayzz-p1011", "eprint_id": 107157, "eprint_status": "archive", "datestamp": "2023-08-22 09:15:54", "lastmod": "2023-10-23 15:35:52", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" }, "orcid": "0000-0002-8380-0921" } ] }, "title": "Walter's basic theorem for fusion systems", "ispublished": "pub", "full_text_status": "restricted", "keywords": "Finite simple groups; Fusion systems", "note": "\u00a9 2020 Elsevier. \n\nReceived 13 September 2018, Available online 8 December 2020. \n\nCommunicated by Markus Linckelmann. \n\nThis work was partially supported by DMS NSF-1265587 and DMS NSF-1601063.", "abstract": "This is the first of two papers determining the saturated 2-fusion systems in which the centralizer of some fully centralized involution contains a component that is the 2-fusion system of a large group of Lie type over a field of odd order.", "date": "2021-03-15", "date_type": "published", "publication": "Journal of Algebra", "volume": "570", "publisher": "Elsevier", "pagerange": "595-610", "id_number": "CaltechAUTHORS:20201217-124921798", "issn": "0021-8693", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20201217-124921798", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1265587" }, { "agency": "NSF", "grant_number": "DMS-1601063" } ] }, "collection": "CaltechAUTHORS", "doi": "10.1016/j.jalgebra.2020.11.018", "resource_type": "article", "pub_year": "2021", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/cny0p-qk635", "eprint_id": 103231, "eprint_status": "archive", "datestamp": "2023-08-20 00:30:55", "lastmod": "2023-10-20 15:48:24", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" }, "orcid": "0000-0002-8380-0921" } ] }, "title": "Fusion systems with alternating J\u2010components", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 2020 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence. \n\nIssue Online: 03 December 2020; Version of Record online: 14 May 2020; Manuscript revised: 10 February 2020; Manuscript received: 05 June 2019. \n\nThis work was partially supported by DMS NSF-1265587 and DMS NSF-1601063.", "abstract": "We essentially determine the saturated 2\u2010fusion systems of J\u2010component type in which the centralizer of some fully centralized involution of maximal 2\u2010rank contains a component that is the 2\u2010fusion system of an alternating group A_n for some n \u2a7e 8.", "date": "2020-12", "date_type": "published", "publication": "Journal of the London Mathematical Society", "volume": "102", "number": "3", "publisher": "London Mathematical Society", "pagerange": "905-956", "id_number": "CaltechAUTHORS:20200515-104035073", "issn": "0024-6107", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200515-104035073", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1265587" }, { "agency": "NSF", "grant_number": "DMS-1601063" } ] }, "doi": "10.1112/jlms.12335", "resource_type": "article", "pub_year": "2020", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/c2t20-f0d81", "eprint_id": 98102, "eprint_status": "archive", "datestamp": "2023-08-22 07:09:02", "lastmod": "2023-10-18 17:03:11", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "The 2-fusion system of an almost simple group", "ispublished": "pub", "full_text_status": "public", "keywords": "Finite groups; Fusion systems", "note": "\u00a9 2019 Elsevier Inc. \n\nReceived 11 February 2019, Available online 22 August 2019. \n\nThis work was partially supported by DMS NSF-1265587 and DMS NSF-1601063.", "abstract": "We show that, under suitable local constraints, the 2-fusion system of an almost simple finite group is almost simple.", "date": "2020-11-01", "date_type": "published", "publication": "Journal of Algebra", "volume": "561", "publisher": "Elsevier", "pagerange": "5-16", "id_number": "CaltechAUTHORS:20190822-100100397", "issn": "0021-8693", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190822-100100397", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1265587" }, { "agency": "NSF", "grant_number": "DMS-1601063" } ] }, "doi": "10.1016/j.jalgebra.2019.08.017", "resource_type": "article", "pub_year": "2020", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/hv2fa-w5m79", "eprint_id": 81892, "eprint_status": "archive", "datestamp": "2023-08-19 00:42:20", "lastmod": "2024-01-14 05:38:19", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "The subgroup structure of finite groups", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2017 American Mathematical Society.", "abstract": "[No abstract]", "date": "2017", "date_type": "published", "publisher": "American Mathematical Society", "place_of_pub": "Providence, RI", "pagerange": "111-121", "id_number": "CaltechAUTHORS:20170928-081918166", "isbn": "978-1-4704-3678-0", "book_title": "Finite Simple Groups: Thirty Years of the Atlas and Beyond", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170928-081918166", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1265587" }, { "agency": "NSF", "grant_number": "DMS-1601063" } ] }, "contributors": { "items": [ { "id": "Bhargava-M", "name": { "family": "Bhargava", "given": "Manjul" } }, { "id": "Guralnick-R", "name": { "family": "Guralnick", "given": "Robert" } }, { "id": "Hiss-G", "name": { "family": "Hiss", "given": "Gerhard" } }, { "id": "Lux-K", "name": { "family": "Lux", "given": "Klaus" } }, { "id": "Tiep-Pham-Huu", "name": { "family": "Tiep", "given": "Pham Huu" } } ] }, "doi": "10.1090/conm/694/13959", "resource_type": "book_section", "pub_year": "2017", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/dfevp-kh744", "eprint_id": 72057, "eprint_status": "archive", "datestamp": "2023-08-20 12:30:30", "lastmod": "2023-10-23 17:49:44", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Oliver-B", "name": { "family": "Oliver", "given": "Bob" } } ] }, "title": "Fusion systems", "ispublished": "pub", "full_text_status": "public", "keywords": "Fusion, Sylow subgroups, finite simple groups, classifying spaces,\nmodular representation theory", "note": "\u00a9 2016 American Mathematical Society. \n\nReceived by the editors December 27, 2015. Published electronically: June 29, 2016. \n\nThe first author was partially supported by NSF DMS-1265587 and NSF DMS-0969009. \n\nThe second author was partially supported by UMR 7539 of the CNRS.\n\n
Published - S0273-0979-2016-01538-2.pdf
", "abstract": "This is a survey article on the theory of fusion systems, a relatively new area of mathematics with connections to local finite group theory, algebraic topology, and modular representation theory. We first describe the general theory and then look separately at these connections.", "date": "2016-06-29", "date_type": "published", "publication": "Bulletin of the American Mathematical Society", "volume": "53", "number": "4", "publisher": "American Mathematical Society", "pagerange": "555-615", "id_number": "CaltechAUTHORS:20161116-124215563", "issn": "0273-0979", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161116-124215563", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1265587" }, { "agency": "NSF", "grant_number": "DMS-0969009" }, { "agency": "Centre National de la Recherche Scientifique (CNRS)", "grant_number": "UMR 7539" } ] }, "doi": "10.1090/bull/1538", "primary_object": { "basename": "S0273-0979-2016-01538-2.pdf", "url": "https://authors.library.caltech.edu/records/dfevp-kh744/files/S0273-0979-2016-01538-2.pdf" }, "resource_type": "article", "pub_year": "2016", "author_list": "Aschbacher, Michael and Oliver, Bob" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/twt7w-tq484", "eprint_id": 67426, "eprint_status": "archive", "datestamp": "2023-08-22 17:33:09", "lastmod": "2023-10-18 21:10:48", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "N-groups and fusion systems", "ispublished": "pub", "full_text_status": "public", "keywords": "Fusion systems; Finite groups", "note": "\u00a9 2015 Elsevier Inc. \n\nThis work was partially supported by NSF grants DMS-1265587 and DMS-0969009.", "abstract": "We classify all saturated 2-fusion systems that are N-systems: that is those systems all of whose local subsystems are solvable, subject to one of the two possible notions of solvability. We also use the result on fusion systems to give a new proof of Thompson's theorem on N-groups; indeed we give a new proof of the theorem determining all finite groups in which all 2-locals are solvable.", "date": "2016-03-01", "date_type": "published", "publication": "Journal of Algebra", "volume": "449", "publisher": "Elsevier", "pagerange": "264-320", "id_number": "CaltechAUTHORS:20160527-090500481", "issn": "0021-8693", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160527-090500481", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-1265587" }, { "agency": "NSF", "grant_number": "DMS-0969009" } ] }, "doi": "10.1016/j.jalgebra.2015.10.011", "resource_type": "article", "pub_year": "2016", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/1dh85-bt711", "eprint_id": 68904, "eprint_status": "archive", "datestamp": "2023-08-20 09:35:28", "lastmod": "2023-10-20 16:18:53", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Daniel Gorenstein, 1923-1992 - A Biographical Memoir by\n Michael Aschbacher", "ispublished": "pub", "full_text_status": "public", "note": "\u00a9 2016 National Academy of Sciences.\n\nPublished - gorenstein-daniel.pdf
", "abstract": "Daniel Gorenstein was one of the most influential figures\nin mathematics during the last few decades of the 20th\ncentury. In particular, he was a primary architect of the\nclassification of the finite simple groups. \n\nDuring his career Gorenstein received many of the honors\nthat the mathematical community reserves for its highest\nachievers. He was awarded the Steele Prize for mathematical\nexposition by the American Mathematical Society in\n1989; he delivered the plenary address at the International\nCongress of Mathematicians in Helsinki, Finland, in 1978;\nand he was the Colloquium Lecturer for the American\nMathematical Society in 1984. He was also a member of\nthe National Academy of Sciences and of the American\nAcademy of Arts and Sciences. \n\nGorenstein was the Jacqueline B. Lewis Professor of\nMathematics at Rutgers University and the founding director of its Center for Discrete\nMathematics and Theoretical Computer Science. He served as chairman of the university's\nmathematics department from 1975 to 1982, and together with his predecessor, Ken\nWolfson, he oversaw a dramatic improvement in the quality of mathematics at Rutgers.", "date": "2016", "date_type": "published", "publication": "Biographical Memoirs", "publisher": "National Academy of Sciences", "pagerange": "1-17", "id_number": "CaltechAUTHORS:20160708-073018195", "issn": "0077-2933", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160708-073018195", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "primary_object": { "basename": "gorenstein-daniel.pdf", "url": "https://authors.library.caltech.edu/records/1dh85-bt711/files/gorenstein-daniel.pdf" }, "resource_type": "article", "pub_year": "2016", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/yzaqr-6fd02", "eprint_id": 66460, "eprint_status": "archive", "datestamp": "2023-08-20 09:34:48", "lastmod": "2024-01-13 16:49:42", "type": "book", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Overgroups of Root Groups in Classical Groups", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Finite groups, linear groups", "note": "\u00a9 2015 American Mathematical Society. \n\nPublished electronically: December 10, 2015.", "abstract": "We extend results of McLaughlin and Kantor on overgroups of long root subgroups and long root elements in finite classical groups. In particular we determine the maximal subgroups of this form. We also determine the maximal overgroups of short root subgroups in finite classical groups, and the maximal overgroups in finite orthogonal groups of c-root subgroups.", "date": "2016", "date_type": "published", "publisher": "American Mathematical Society", "place_of_pub": "Providence, RI", "id_number": "CaltechAUTHORS:20160425-141046633", "isbn": "978-1-4704-1845-8", "book_title": "Overgroups of root groups in classical groups", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160425-141046633", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "contributors": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "doi": "10.1090/memo/1140", "resource_type": "book", "pub_year": "2016", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/5af6y-91a67", "eprint_id": 42897, "eprint_status": "archive", "datestamp": "2023-08-19 22:50:52", "lastmod": "2023-10-25 23:08:10", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Finite Groups of Seitz Type", "ispublished": "pub", "full_text_status": "public", "note": "\u00a9 2013 American Mathematical Society \nThe copyright for this article reverts to public domain 28 years after publication.\n\nReceived by editor(s): February 23, 2012;\nReceived by editor(s) in revised form: March 7, 2012, and March 9, 2012; \nPosted: October 4, 2013.\n\n\nThis work was partially supported by NSF grants DMS-0504852 and DMS-0969009.\n\nPublished - S0002-9939-2013-11752-1.pdf
", "abstract": "We show that a useful condition of Seitz on finite groups of Lie\ntype over fields of order q > 4 is often satisfied when q is 2 or 3. We also\nobserve that various consequences of the Seitz condition, established by Seitz\nand Cline, Parshall, and Scott when q > 4, also hold when q is 3 or 4.", "date": "2014-01", "date_type": "published", "publication": "Proceedings of the American Mathematical Society", "volume": "142", "number": "1", "publisher": "American Mathematical Society", "pagerange": "113-120", "id_number": "CaltechAUTHORS:20131209-104216789", "issn": "0002-9939", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20131209-104216789", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0504852" }, { "agency": "NSF", "grant_number": "DMS-0969009" } ] }, "other_numbering_system": { "items": [ { "id": "MR3119186", "name": "MathSciNet Review" } ] }, "doi": "10.1090/S0002-9939-2013-11752-1", "primary_object": { "basename": "S0002-9939-2013-11752-1.pdf", "url": "https://authors.library.caltech.edu/records/5af6y-91a67/files/S0002-9939-2013-11752-1.pdf" }, "resource_type": "article", "pub_year": "2014", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/1f057-24g21", "eprint_id": 38378, "eprint_status": "archive", "datestamp": "2023-08-22 09:22:22", "lastmod": "2023-10-23 20:06:48", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Overgroup lattices in finite groups of Lie type containing a parabolic", "ispublished": "pub", "full_text_status": "restricted", "keywords": "Finite groups; Groups of Lie type; Subgroup lattices", "note": "\u00a9 2013 Elsevier Inc. \nReceived 26 June 2012;\nAvailable online 6 March 2013.\nCommunicated by Leonard L. Scott, Jr.\n\nThis work was partially supported by NSF grants DMS-0504852 and DMS-0969009.", "abstract": "The main theorem is a step in a program to show there exist finite lattices that are not an interval in the lattice of subgroups of any finite group. As part of the proof of the main theorem, we prove a theorem on the structure of maximal parabolics in finite groups of Lie type, which is of independent interest.", "date": "2013-05-15", "date_type": "published", "publication": "Journal of Algebra", "volume": "382", "publisher": "Elsevier", "pagerange": "71-99", "id_number": "CaltechAUTHORS:20130509-100832818", "issn": "0021-8693", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130509-100832818", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0504852" }, { "agency": "NSF", "grant_number": "DMS-0969009" } ] }, "doi": "10.1016/j.jalgebra.2013.01.034", "resource_type": "article", "pub_year": "2013", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/npe69-kz005", "eprint_id": 37651, "eprint_status": "archive", "datestamp": "2023-08-22 08:55:31", "lastmod": "2023-10-23 17:54:23", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Fusion systems of F_2-type", "ispublished": "pub", "full_text_status": "restricted", "keywords": "Fusion systems; Finite groups", "note": "\u00a9 2013 Elsevier Inc. Received 19 January 2012. Available online 24 January 2013. Communicated by Ronald Solomon. This work was partially supported by NSF grants DMS-0504852 and DMS-0969009.", "abstract": "We prove results on 2-fusion systems related to the 2-fusion systems of groups of Lie type over the field of order 2 and certain sporadic groups. The results are used in a later paper to determine the N-systems: the 2-fusion systems of N-groups.", "date": "2013-03-15", "date_type": "published", "publication": "Journal of Algebra", "volume": "378", "publisher": "Elsevier", "pagerange": "217-262", "id_number": "CaltechAUTHORS:20130327-114824399", "issn": "0021-8693", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130327-114824399", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0504852" }, { "agency": "NSF", "grant_number": "DMS-0969009" } ] }, "doi": "10.1016/j.jalgebra.2012.12.018", "resource_type": "article", "pub_year": "2013", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/tecsz-6kh56", "eprint_id": 38848, "eprint_status": "archive", "datestamp": "2023-08-22 08:36:48", "lastmod": "2023-10-23 23:32:09", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "S_3-free 2-fusion systems", "ispublished": "pub", "full_text_status": "public", "keywords": "finite groups; fusion systems; Sylow subgroups", "note": "\u00a9 2012 Edinburgh Mathematical Society.\n\nPublished online: 05 December 2012.\n\nThis work was partly supported by NSF Grants DMS-0504852\nand DMS-0969009. The author thanks the referee for a number of suggestions leading to\nimprovements in the paper.\n\nPublished - Aschbacher_2013p27.pdf
", "abstract": "We develop a theory of 2-fusion systems of even characteristic, and use that theory to show that all S_3-free saturated 2-fusion systems are constrained. This supplies a new proof of Glauberman's Theorem on S_4-free groups and its various corollaries.", "date": "2013-02", "date_type": "published", "publication": "Proceedings of the Edinburgh Mathematical Society", "volume": "56", "number": "1", "publisher": "Edinburgh Mathematical Society", "pagerange": "27-48", "id_number": "CaltechAUTHORS:20130607-083326320", "issn": "0013-0915", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130607-083326320", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0504852" }, { "agency": "NSF", "grant_number": "DMS-0969009" } ] }, "other_numbering_system": { "items": [ { "id": "MR3021403", "name": "MathSciNet Review" } ] }, "doi": "10.1017/S0013091512000235", "primary_object": { "basename": "Aschbacher_2013p27.pdf", "url": "https://authors.library.caltech.edu/records/tecsz-6kh56/files/Aschbacher_2013p27.pdf" }, "resource_type": "article", "pub_year": "2013", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/td2h0-tae78", "eprint_id": 88631, "eprint_status": "archive", "datestamp": "2023-08-19 10:34:20", "lastmod": "2023-10-18 22:11:56", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Lyons-R", "name": { "family": "Lyons", "given": "Richard" } }, { "id": "Smith-S", "name": { "family": "Smith", "given": "Steve" } }, { "id": "Solomon-R", "name": { "family": "Solomon", "given": "Ronald" } } ] }, "title": "2012 Steele Prizes", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 2012 American Mathematical Society.", "abstract": "The 2012 Leroy P. Steele Prize for Mathematical\nExposition is awarded to Michael Aschbacher,\nRichard Lyons, Steve Smith, and Ronald Solomon\nfor their work, The Classification of Finite Simple\nGroups: Groups of Characteristic 2 Type, Mathematical\nSurveys and Monographs, 172, American\nMathematical Society, Providence, RI, 2011. In this\npaper, the authors, who have done foundational\nwork in the classification of finite simple groups,\noffer to the general mathematical public an articulate\nand readable exposition of the classification\nof characteristic 2 type groups.", "date": "2012-04", "date_type": "published", "publication": "Notices of the American Mathematical Society", "volume": "59", "number": "4", "publisher": "American Mathematical Society", "pagerange": "563-566", "id_number": "CaltechAUTHORS:20180807-133142929", "issn": "0002-9920", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180807-133142929", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.1090/noti826", "resource_type": "article", "pub_year": "2012", "author_list": "Aschbacher, Michael; Lyons, Richard; et el." }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/9saf3-dsg11", "eprint_id": 31651, "eprint_status": "archive", "datestamp": "2023-08-19 10:07:40", "lastmod": "2023-10-17 18:44:59", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Lower signalizer lattices in alternating and symmetric groups", "ispublished": "pub", "full_text_status": "public", "note": "\u00a9 2012 de Gruyter.\n\nReceived September 18, 2010; revised November 28, 2011.\nPublished Online: 06/03/2012.\nCommunicated by Robert M. Guralnick.\nThis work was partially supported by NSF DMS-0504852 and NSF DMS-0969009.\n\nPublished - Aschbacher2012p18270J_Group_Theory.pdf
", "abstract": "We prove that the subgroup lattices of finite alternating and symmetric groups\ndo not contain so-called lower signalizer lattices in the class D\ufffd\u0394. This result is one step\nin a program to show that the lattices in the class D\ufffd\u0394 are not isomorphic to an interval in\nthe subgroup lattice of any finite group.", "date": "2012-03", "date_type": "published", "publication": "Journal of Group Theory", "volume": "15", "number": "2", "publisher": "Walter de Gruyter", "pagerange": "151-225", "id_number": "CaltechAUTHORS:20120525-104153526", "issn": "1433-5883", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120525-104153526", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0504852" }, { "agency": "NSF", "grant_number": "DMS-0969009" } ] }, "doi": "10.1515/jgt-2011-0112", "primary_object": { "basename": "Aschbacher2012p18270J_Group_Theory.pdf", "url": "https://authors.library.caltech.edu/records/9saf3-dsg11/files/Aschbacher2012p18270J_Group_Theory.pdf" }, "resource_type": "article", "pub_year": "2012", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/r5359-bqa87", "eprint_id": 88625, "eprint_status": "archive", "datestamp": "2023-08-19 05:02:50", "lastmod": "2024-01-14 20:36:42", "type": "book", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Kessar-R", "name": { "family": "Kessar", "given": "Radha" } }, { "id": "Oliver-B", "name": { "family": "Oliver", "given": "Bob" } } ] }, "title": "Fusion Systems in Algebra and Topology", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2011 Cambridge University Press.", "abstract": "A fusion system over a p-group S is a category whose objects form the set of all subgroups of S, whose morphisms are certain injective group homomorphisms, and which satisfies axioms first formulated by Puig that are modelled on conjugacy relations in finite groups. The definition was originally motivated by representation theory, but fusion systems also have applications to local group theory and to homotopy theory. The connection with homotopy theory arises through classifying spaces which can be associated to fusion systems and which have many of the nice properties of p-completed classifying spaces of finite groups. Beginning with a detailed exposition of the foundational material, the authors then proceed to discuss the role of fusion systems in local finite group theory, homotopy theory and modular representation theory. This book serves as a basic reference and as an introduction to the field, particularly for students and other young mathematicians.", "date": "2011", "date_type": "published", "publisher": "Cambridge University Press", "place_of_pub": "Cambridge", "id_number": "CaltechAUTHORS:20180807-124803752", "isbn": "9781139003841", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180807-124803752", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.1017/CBO9781139003841", "resource_type": "book", "pub_year": "2011", "author_list": "Aschbacher, Michael; Kessar, Radha; et el." }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/bgvds-kvv30", "eprint_id": 18302, "eprint_status": "archive", "datestamp": "2023-08-19 02:18:54", "lastmod": "2023-10-20 15:57:30", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Generation of fusion systems of characteristic 2-type", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 2009 Springer-Verlag. \n\nReceived: 9 December 2008. Accepted: 25 November 2009. Published online: 11 December 2009. \n\nThis work was partially supported by NSF-0504852.", "abstract": "We prove that if F is a saturated fusion system on a finite 2-\ngroup S, then either F is known, or F is generated by the normalizers of\ntwo canonically defined F-characteristic subgroups of S. There are various\ncorollaries for finite groups of characteristic 2-type.", "date": "2010-05", "date_type": "published", "publication": "Inventiones Mathematicae", "volume": "180", "number": "2", "publisher": "Springer", "pagerange": "225-299", "id_number": "CaltechAUTHORS:20100513-152930353", "issn": "0020-9910", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100513-152930353", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0504852" } ] }, "doi": "10.1007/s00222-009-0229-z", "resource_type": "article", "pub_year": "2010", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/qta7e-y5811", "eprint_id": 19315, "eprint_status": "archive", "datestamp": "2023-08-19 01:50:13", "lastmod": "2023-10-20 20:32:51", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Chermak-A", "name": { "family": "Chermak", "given": "Andrew" } } ] }, "title": "A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 2009 Annals of Mathematics.\nReceived January 12, 2006.\nRevised December 14, 2006.\nThe work of the first author was partially supported by NSF-0203417.", "abstract": "We extend the notion of a p-local finite group (defined in [BLO03]) to the notion of a p-local group. We define morphisms of p-local groups, obtaining thereby a category, and we introduce the notion of a representation of a p-local group via signalizer functors associated with groups. We construct a chain G = (G_0 \u2192 G_1 \u2192 ...) of 2-local finite groups, via a representation of a chain G^* = (G_0 \u2192 G_1 \u2192 ...) of groups, such that G_0 is the 2-local finite group of the third Conway sporadic group Co_3, and for n > 0, G_n is one of the 2-local finite groups constructed by Levi and Oliver in [LO02]. We show that the direct limit G of G exists in the category of 2-local groups, and that it is the 2-local group of the union of the chain G^*. The 2-completed classifying space of G is shown to be the classifying space B DI(4) of the exotic 2-compact group of Dwyer and Wilkerson [DW93].", "date": "2010-03", "date_type": "published", "publication": "Annals of Mathematics", "volume": "171", "number": "2", "publisher": "Annals of Mathematics", "pagerange": "881-978", "id_number": "CaltechAUTHORS:20100806-092914498", "issn": "0003-486X", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100806-092914498", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "0203417" } ] }, "collection": "CaltechAUTHORS", "resource_type": "article", "pub_year": "2010", "author_list": "Aschbacher, Michael and Chermak, Andrew" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/be14c-2nz37", "eprint_id": 16939, "eprint_status": "archive", "datestamp": "2023-08-21 22:28:53", "lastmod": "2023-10-19 22:40:30", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Shareshian-J", "name": { "family": "Shareshian", "given": "John" } } ] }, "title": "Restrictions on the structure of subgroup lattices of finite alternating and symmetric groups", "ispublished": "pub", "full_text_status": "restricted", "keywords": "Subgroup lattice; Symmetric group; Alternating group", "note": "\u00a9 2009 Elsevier Inc. \n\nReceived 9 January 2009. Available online 23 July 2009. \n\nCommunicated by Ronald Solomon. \n\nWe thank the referee for helpful comments. We thank Russ Woodroofe for providing the picture of D\u0394(3, 3).\n\nPartially supported by NSF Grants DMS 0504852 and DMS 0604233.", "abstract": "Let G be a finite alternating or symmetric group. We describe an infinite class of finite lattices, none of which is isomorphic to any interval [H,G] in the subgroup lattice of G.", "date": "2009-10-01", "date_type": "published", "publication": "Journal of Algebra", "volume": "322", "number": "7", "publisher": "Elsevier", "pagerange": "2449-2463", "id_number": "CaltechAUTHORS:20091210-094932243", "issn": "0021-8693", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20091210-094932243", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0504852" }, { "agency": "NSF", "grant_number": "DMS-0604233" } ] }, "doi": "10.1016/j.jalgebra.2009.05.042", "resource_type": "article", "pub_year": "2009", "author_list": "Aschbacher, Michael and Shareshian, John" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/7kxjk-xhy82", "eprint_id": 15117, "eprint_status": "archive", "datestamp": "2023-08-21 22:16:31", "lastmod": "2023-10-18 20:59:36", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Overgroups of primitive groups, II", "ispublished": "pub", "full_text_status": "restricted", "keywords": "Finite groups; Permutation groups", "note": "\u00a9 2009 Elsevier. \n\nReceived 19 August 2008. Communicated by Ronald Solomon. Available online 30 May 2009.", "abstract": "We continue our study of the overgroup lattices of subgroups of finite alternating and symmetric groups, with applications to the question of Palfy and Pudlak as to whether each finite lattice is an interval in the lattice of subgroups of some finite group.", "date": "2009-09-01", "date_type": "published", "publication": "Journal of Algebra", "volume": "322", "number": "5", "publisher": "Elsevier", "pagerange": "1586-1626", "id_number": "CaltechAUTHORS:20090817-144817879", "issn": "0021-8693", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20090817-144817879", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.1016/j.jalgebra.2009.04.044", "resource_type": "article", "pub_year": "2009", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/3m2gh-ec774", "eprint_id": 16403, "eprint_status": "archive", "datestamp": "2023-08-21 21:58:19", "lastmod": "2023-10-19 22:11:44", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Overgroups of Primitive Groups", "ispublished": "pub", "full_text_status": "restricted", "keywords": "permutation group; lattice", "note": "\u00a9 2009 Australian Mathematical Publishing Association Inc. \n\nReceived September 1 2007, accepted June 6 2008. \n\nThis work was partially supported by grant no. NSF-0504852.", "abstract": "We give a qualitative description of the set O_G(H) of overgroups in G of primitive subgroups H of finite\nalternating and symmetric groups G, and particularly of the maximal overgroups. We then show that\ncertain weak restrictions on the lattice O_G(H) impose strong restrictions on H and its overgroup lattice.", "date": "2009-08", "date_type": "published", "publication": "Journal of the Australian Mathematical Sociey", "volume": "87", "number": "1", "publisher": "Australian Mathematical Society", "pagerange": "37-82", "id_number": "CaltechAUTHORS:20091020-133518121", "issn": "1446-7887", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20091020-133518121", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0504852" } ] }, "doi": "10.1017/S1446788708000785", "resource_type": "article", "pub_year": "2009", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/p2hzh-8r107", "eprint_id": 14942, "eprint_status": "archive", "datestamp": "2023-08-20 00:16:28", "lastmod": "2023-10-18 20:23:55", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Signalizer lattices in finite groups", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 2009 The University of Michigan.\n\nReceived February 12, 2007. Revision received January 31, 2008.\nThis work was partially supported by NSF-0504852.\nZentralblatt MATH identifier: 05566074.", "abstract": "Let G be a finite group and let H be a subgroup of G. We investigate constraints\nimposed upon the structure of G by restrictions on the lattice O_G(H) of overgroups\nof H in G. Call such a lattice a finite group interval lattice. In particular\nwe would like to show that the following question has a positive answer.", "date": "2009", "date_type": "published", "publication": "Michigan Mathematical Journal", "volume": "58", "number": "1", "publisher": "University of Michigan", "pagerange": "79-103", "id_number": "CaltechAUTHORS:20090811-091248605", "issn": "0026-2285", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20090811-091248605", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "0504852" } ] }, "doi": "10.1307/mmj/1242071684", "resource_type": "article", "pub_year": "2009", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/gst1n-3qv43", "eprint_id": 88535, "eprint_status": "archive", "datestamp": "2023-08-19 23:24:20", "lastmod": "2023-10-18 22:07:33", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "On a question of Farjoun", "ispublished": "unpub", "full_text_status": "public", "note": "\u00a9 2008 Walter de Gruyter GmbH.", "abstract": "[no abstract]", "date": "2008-08", "date_type": "published", "publisher": "De Gruyter", "place_of_pub": "Berlin", "pagerange": "1-28", "id_number": "CaltechAUTHORS:20180802-143851994", "isbn": "978-3-11-019812-6", "book_title": "Finite Groups 2003 : Proceedings of the Gainesville Conference on Finite Groups", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180802-143851994", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "contributors": { "items": [ { "id": "Ho-Chat-Yin", "name": { "family": "Ho", "given": "Chat Yin" } }, { "id": "Sin-Peter", "name": { "family": "Sin", "given": "Peter" } }, { "id": "Tiep-P-H", "name": { "family": "Tiep", "given": "Pham Huu" } }, { "id": "Turull-A", "name": { "family": "Turull", "given": "Alexandre" } } ] }, "doi": "10.1515/9783110198126.1", "resource_type": "book_section", "pub_year": "2008", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/s19mn-5r963", "eprint_id": 88733, "eprint_status": "archive", "datestamp": "2023-08-19 23:05:26", "lastmod": "2023-10-18 22:16:26", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Normal subsystems of fusion systems", "ispublished": "pub", "full_text_status": "public", "note": "\u00a9 2008 London Mathematical Society. \n\nIssue Online: 21 March 2008; Version of Record online: 21 March 2008; Manuscript revised: 16 August 2007; Manuscript received: 14 November 2006. \n\nThis work was partially supported by NSF\u20100504852.", "abstract": "The notion of a fusion system was first defined and explored by Puig in the context of modular representation theory. Later, Broto, Levi, and Oliver significantly extended the theory of fusion systems as a tool in homotopy theory. In this paper we begin a program to establish a local theory of fusion systems similar to the local theory of finite groups. In particular, we define the notion of a normal subsystem of a saturated fusion system, and prove some basic results about normal subsystems and factor systems.", "date": "2008-07", "date_type": "published", "publication": "Proceedings of the London Mathematical Society", "volume": "97", "number": "1", "publisher": "London Mathematical Society", "pagerange": "239-271", "id_number": "CaltechAUTHORS:20180810-075919663", "issn": "0024-6115", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-075919663", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0504852" } ] }, "doi": "10.1112/plms/pdm057", "resource_type": "article", "pub_year": "2008", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/cm5e5-ptt87", "eprint_id": 13447, "eprint_status": "archive", "datestamp": "2023-08-19 22:23:36", "lastmod": "2023-10-17 23:54:16", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "On intervals in subgroup lattices of finite groups", "ispublished": "pub", "full_text_status": "public", "keywords": "INTERMEDIATE SUBFACTORS.", "note": "\u00a9 2008 American Mathematical Society. Received by the editors June 28, 2006. Article electronically published on March 17, 2008. This work was partially supported by NSF-0504852. 20D30, 06B05, 46L37\n\nPublished - ASCjams08.pdf
", "abstract": "We investigate the question of which finite lattices L are isomorphic to the lattice [H,G] of all overgroups of a subgroup H in a finite group G. We show that the structure of G is highly restricted if [H,G] is disconnected. We define the notion of a \"signalizer lattice\" in H and show for suitable disconnected lattices L, if [H,G] is minimal subject to being isomorphic to L or its dual, then either G is almost simple or H admits a signalizer lattice isomorphic to L or its dual. We use this theory to answer a question in functional analysis raised by Watatani.", "date": "2008-03-17", "date_type": "published", "publication": "Journal of the American Mathematical Society", "volume": "21", "number": "3", "publisher": "American Mathematical Society", "pagerange": "809-830", "id_number": "CaltechAUTHORS:ASCjams08", "issn": "0894-0347", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:ASCjams08", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "0504852" } ] }, "doi": "10.1090/S0894-0347-08-00602-4", "primary_object": { "basename": "ASCjams08.pdf", "url": "https://authors.library.caltech.edu/records/cm5e5-ptt87/files/ASCjams08.pdf" }, "resource_type": "article", "pub_year": "2008", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/h0k3e-2fp33", "eprint_id": 98246, "eprint_status": "archive", "datestamp": "2023-08-19 19:48:41", "lastmod": "2023-10-18 17:11:37", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Childs-A-M", "name": { "family": "Childs", "given": "Andrew M." } }, { "id": "Wocjan-P", "name": { "family": "Wocjan", "given": "Pawe\u0142" } } ] }, "title": "The limitations of nice mutually unbiased bases", "ispublished": "pub", "full_text_status": "public", "keywords": "Quantum information theory . Mutually unbiased bases . Quantum designs", "note": "\u00a9 Springer Science + Business Media, LLC 2006. \n\nPublished online: 11 July 2006. \n\nMA is supported by the National Science Foundation under Grant No. DMS-0203417. AMC and PW are supported by the National Science Foundation under Grant No. EIA-0086038.\n\nSubmitted - 0412066.pdf
", "abstract": "Mutually unbiased bases of a Hilbert space can be constructed by partitioning a unitary error basis. We consider this construction when the unitary error basis is a nice error basis. We show that the number of resulting mutually unbiased bases can be at most one plus the smallest prime power contained in the dimension, and therefore that this construction cannot improve upon previous approaches. We prove this by establishing a correspondence between nice mutually unbiased bases and abelian subgroups of the index group of a nice error basis and then bounding the number of such subgroups. This bound also has implications for the construction of certain combinatorial objects called nets.", "date": "2007-03", "date_type": "published", "publication": "Journal of Algebraic Combinatorics", "volume": "25", "number": "2", "publisher": "Springer", "pagerange": "111-123", "id_number": "CaltechAUTHORS:20190826-124740760", "issn": "0925-9899", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190826-124740760", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0203417" }, { "agency": "NSF", "grant_number": "EIA-0086038" } ] }, "doi": "10.1007/s10801-006-0002-y", "primary_object": { "basename": "0412066.pdf", "url": "https://authors.library.caltech.edu/records/h0k3e-2fp33/files/0412066.pdf" }, "resource_type": "article", "pub_year": "2007", "author_list": "Aschbacher, Michael; Childs, Andrew M.; et el." }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/47bvp-rpt47", "eprint_id": 18097, "eprint_status": "archive", "datestamp": "2023-08-22 07:55:47", "lastmod": "2023-10-20 15:35:06", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Guralnick-R", "name": { "family": "Guralnick", "given": "Robert" } }, { "id": "Segev-Y", "name": { "family": "Segev", "given": "Yoav" } } ] }, "title": "Elementary abelian 2-subgroups of Sidki-type in finite groups", "ispublished": "pub", "full_text_status": "restricted", "keywords": "finite simple groups; involutions; parabolic subgroups; fundamental subgroups; saturation", "note": "\u00a9 2007 European Mathematical Society.\nReceived November 3, 2006; revised March 6, 2007.\nPartially supported by NSF-0504852.\nPartially supported by NSF-0140578.\nPartially supported by BSF grant no. 2004-083.\nThe referee report consisted of six pages of detailed, useful comments.\nWe thank and applaud the referee for this work.", "abstract": "Let G be a finite group. We say that a nontrivial elementary abelian 2-subgroup V of\nG is of Sidki-type in G, if for each involution i in G, C_V(i) \u2260 1. A conjecture due to S. Sidki\n(J. Algebra 39, 1976) asserts that if V is of Sidki-type in G, then V \u2229 0_2(G) \u2260 1. In this paper\nwe prove a stronger version of Sidki's conjecture. As part of the proof, we also establish weak\nversions of the saturation results of G. Seitz (Invent. Math. 141, 2000) for involutions in finite\ngroups of Lie type in characteristic 2. Seitz's results apply to elements of order p in groups\nof Lie type in characteristic p, but only when p is a good prime, and 2 is usually not a good\nprime.", "date": "2007", "date_type": "published", "publication": "Groups, Geometry, and Dynamics", "volume": "1", "number": "4", "publisher": "European Mathematical Society", "pagerange": "347-400", "id_number": "CaltechAUTHORS:20100503-094555816", "issn": "1661-7207", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100503-094555816", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "0504852" }, { "agency": "NSF", "grant_number": "0140578." }, { "agency": "United States-Israel Binational Science Foundation (BSF)", "grant_number": "2004-083" } ] }, "doi": "10.4171/GGD/18", "resource_type": "article", "pub_year": "2007", "author_list": "Aschbacher, Michael; Guralnick, Robert; et el." }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/c4adw-9py07", "eprint_id": 22090, "eprint_status": "archive", "datestamp": "2023-08-19 18:08:05", "lastmod": "2023-10-23 15:40:42", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Kinyon-M-K", "name": { "family": "Kinyon", "given": "Michael K." } }, { "id": "Phillips-J-D", "name": { "family": "Phillips", "given": "J. D." } } ] }, "title": "Finite Bruck loops", "ispublished": "pub", "full_text_status": "public", "note": "\u00a9 2005 American Mathematical Society. \n\nReceived by the editors December 15, 2003 and, in revised form, June 29, 2004. Article electronically published on September 22, 2005. \n\nThe first author was partially supported by NSF-0203417.\n\nPublished - ASCtams06.pdf
Submitted - 0401193.pdf
", "abstract": "Bruck loops are Bol loops satisfying the automorphic inverse property. We prove a structure theorem for finite Bruck loops X, showing that X is essentially the direct product of a Bruck loop of odd order with a 2-element Bruck loop. The former class of loops is well understood. We identify the minimal obstructions to the conjecture that all finite 2-element Bruck loops are 2-loops, leaving open the question of whether such obstructions actually exist.", "date": "2006-07", "date_type": "published", "publication": "Transactions of the American Mathematical Society", "volume": "358", "number": "7", "publisher": "American Mathematical Society", "pagerange": "3061-3075", "id_number": "CaltechAUTHORS:20110209-094820472", "issn": "0002-9947", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110209-094820472", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0203417" } ] }, "doi": "10.1090/S0002-9947-05-03778-5", "primary_object": { "basename": "0401193.pdf", "url": "https://authors.library.caltech.edu/records/c4adw-9py07/files/0401193.pdf" }, "related_objects": [ { "basename": "ASCtams06.pdf", "url": "https://authors.library.caltech.edu/records/c4adw-9py07/files/ASCtams06.pdf" } ], "resource_type": "article", "pub_year": "2006", "author_list": "Aschbacher, Michael; Kinyon, Michael K.; et el." }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/38qww-ejv63", "eprint_id": 105400, "eprint_status": "archive", "datestamp": "2023-08-22 04:19:10", "lastmod": "2023-10-20 21:57:38", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Highly complex proofs and implications of such proofs", "ispublished": "pub", "full_text_status": "restricted", "keywords": "complex; proof; simple group; classification", "note": "\u00a9 2005 The Royal Society. \n\nDiscussion Meeting Issue 'The nature of mathematical proof' organized by A. Bundy, M. Atiyah, A. Macintyre and D. Mackenzie.\n\nThis work was partially supported by NSF-0203417.", "abstract": "Conventional wisdom says the ideal proof should be short, simple, and elegant. However there are now examples of very long, complicated proofs, and as mathematics continues to mature, more examples are likely to appear. Such proofs raise various issues. For example it is impossible to write out a very long and complicated argument without error, so is such a 'proof' really a proof? What conditions make complex proofs necessary, possible, and of interest? Is the mathematics involved in dealing with information rich problems qualitatively different from more traditional mathematics?", "date": "2005-10-15", "date_type": "published", "publication": "Philosophical Transactions A: Mathematical, Physical and Engineering Sciences", "volume": "363", "number": "1835", "publisher": "Royal Society of London", "pagerange": "2401-2406", "id_number": "CaltechAUTHORS:20200916-090615838", "issn": "1364-503X", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200916-090615838", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-0203417" } ] }, "doi": "10.1098/rsta.2005.1655", "resource_type": "article", "pub_year": "2005", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/5b4mm-frv50", "eprint_id": 24909, "eprint_status": "archive", "datestamp": "2023-08-19 13:59:50", "lastmod": "2023-10-24 15:02:20", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "The Status of the Classification of the Finite Simple Groups", "ispublished": "pub", "full_text_status": "public", "note": "\u00a9 2004 American Mathematical Society.\nThis work was partially supported by NSF-0203417.\n\nPublished - ASCnams04.pdf
", "abstract": "The classification of the finite simple groups is one of the great theorems of recent mathematics. One of its principal participants reviews the result and current progress on understanding it.", "date": "2004-08", "date_type": "published", "publication": "Notices of the American Mathematical Society", "volume": "51", "number": "7", "publisher": "American Mathematical Society", "pagerange": "736-740", "id_number": "CaltechAUTHORS:20110817-103540228", "issn": "0002-9920", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110817-103540228", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "0203417" } ] }, "primary_object": { "basename": "ASCnams04.pdf", "url": "https://authors.library.caltech.edu/records/5b4mm-frv50/files/ASCnams04.pdf" }, "resource_type": "article", "pub_year": "2004", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/m3wrn-5w840", "eprint_id": 3654, "eprint_status": "archive", "datestamp": "2023-08-21 23:59:31", "lastmod": "2023-10-16 16:05:03", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "A 2-local characterization of M(12)", "ispublished": "pub", "full_text_status": "public", "note": "\u00a92003 University of Illinois. \n\nReceived August 16, 2002.\nDedicated to Reinhold Baer on the 100th anniversary of his birth. This work was partially supported by NSF DMS-0203417.", "abstract": "A characterization of the Mathieu group M(12) is established; the characterization is used by Aschbacher and Smith in their classification of the quasithin finite simple groups.", "date": "2003", "date_type": "published", "publication": "Illinois Journal of Mathematics", "volume": "47", "number": "1-2", "publisher": "Illinois Journal of Mathematics", "pagerange": "31-47", "id_number": "CaltechAUTHORS:ASCijm03", "issn": "0019-2082", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:ASCijm03", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "primary_object": { "basename": "ASCijm03.pdf", "url": "https://authors.library.caltech.edu/records/m3wrn-5w840/files/ASCijm03.pdf" }, "related_objects": [ { "basename": "aschbacher.ps", "url": "https://authors.library.caltech.edu/records/m3wrn-5w840/files/aschbacher.ps" } ], "resource_type": "article", "pub_year": "2003", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/n5tc0-76369", "eprint_id": 78889, "eprint_status": "archive", "datestamp": "2023-08-19 06:54:21", "lastmod": "2023-10-26 14:26:04", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "On Primitive Linear Representations of Finite Groups", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 2000 Academic Press. \n\nReceived 12 April 2000. \n\nThis work was partially supported by National Science Foundation grant NSF-9901367.", "abstract": "Let F be a field, let G be a finite group, and let \u03c0 be a linear representation of G over F; that is, \u03c0 is a group homomorphism \u03c0: G \u2192 GL(V) of G into the general linear group on a finite-dimensional vector space V over F.\nWe say \u03c0 is AI if \u03c0 is completely reducible and for each normal subgroup H of G, each irreducible FH-submodule of V is absolutely irreducible. For example, if F is algebraically closed then all completely reducible representations over F are AI. In particular, all of our theorems hold over the complex numbers without the hypothesis that the representation is AI.", "date": "2000-12-15", "date_type": "published", "publication": "Journal of Algebra", "volume": "234", "number": "2", "publisher": "Elsevier", "pagerange": "627-640", "id_number": "CaltechAUTHORS:20170710-101117286", "issn": "0021-8693", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170710-101117286", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-9901367" } ] }, "doi": "10.1006/jabr.2000.8532", "resource_type": "article", "pub_year": "2000", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/ytern-gd019", "eprint_id": 103132, "eprint_status": "archive", "datestamp": "2023-08-22 12:28:31", "lastmod": "2024-01-15 03:02:03", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael G." } }, { "id": "Smith-S-D", "name": { "family": "Smith", "given": "Stephen D." } } ] }, "title": "Quasithin Groups", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Finite Group; Weyl Group; Uniqueness Theorem; Finite Simple Group; Subnormal Subgroup", "note": "\u00a9 1998 Birkhiiuser Verlag Basel/Switzerland. \n\nPartially supported by NSF grants DMS-91-01237 and 96-22843. \n\nPartially supported by NSA grant MDA 904-93-H-3039.", "abstract": "Geoff Mason announced in about 1980 the classifcication of quasithin groups of characteristic 2; but never published this step in the classification of the finite simple groups. In January 1996, the authors began work toward a new and more general classification of quasithin groups; the paper gives an exposition of the approach and considerable progress to date.", "date": "1998", "date_type": "published", "publisher": "Springer", "place_of_pub": "Basel", "pagerange": "1-7", "id_number": "CaltechAUTHORS:20200512-100004286", "isbn": "978-3-0348-9785-3", "book_title": "Groups and Geometries", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200512-100004286", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-91-01237" }, { "agency": "NSF", "grant_number": "DMS-96-22843" }, { "agency": "National Security Agency", "grant_number": "MDA 904-93-H-3039" } ] }, "contributors": { "items": [ { "id": "di-Martino-L", "name": { "family": "di Martino", "given": "Lino" } }, { "id": "Kantor-W-M", "name": { "family": "Kantor", "given": "William M." } }, { "id": "Lunardon-G", "name": { "family": "Lunardon", "given": "Guglielmo" } }, { "id": "Pasini-A", "name": { "family": "Pasini", "given": "Antonio" } }, { "id": "Tamburini-M-C", "name": { "family": "Tamburini", "given": "Maria Clara" } } ] }, "doi": "10.1007/978-3-0348-8819-6_1", "resource_type": "book_section", "pub_year": "1998", "author_list": "Aschbacher, Michael G. and Smith, Stephen D." }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/g57yg-2jx07", "eprint_id": 103124, "eprint_status": "archive", "datestamp": "2023-08-22 12:28:23", "lastmod": "2024-01-15 03:01:53", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Quasithin Groups", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Finite Group; Simple Group; Borel Subgroup; Cartan Subgroup; Finite Simple Group", "note": "\u00a9 1998 Springer Science+Business Media Dordrecht. \n\nThis work was partially supported by NSF-9622843.", "abstract": "The treatment of quasithin groups of characteristic 2 was one of the last steps in the Classification of the finite simple groups. Geoff Mason [12] announced a classification of these groups in about 1980, but never published his work. A few people have a copy of a large manuscript containing his efforts, but because it was distributed slowly, section by section, it was only during the last few years that it was realized that Mason's manuscript is incomplete in various ways. A few years ago I wrote up a treatment which begins where Mason's manuscript ends and finishes the problem assuming the results he says he proves. I have only read Mason's manuscript superficially, but it appears there are missing lemmas even for the part of the problem the theorems in his manuscript cover. I do believe however that he has seriously addressed the issues involved and that he could turn his manuscript into a proof with enough work. However Mason is now involved with Moonshine and has no interest in completing or publishing his manuscript.", "date": "1998", "date_type": "published", "publisher": "Springer", "place_of_pub": "Dordrecht", "pagerange": "321-340", "id_number": "CaltechAUTHORS:20200512-081608756", "isbn": "978-0-7923-5292-1", "book_title": "Algebraic Groups and their Representations", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200512-081608756", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-9622843" } ] }, "contributors": { "items": [ { "id": "Carter-R-W", "name": { "family": "Carter", "given": "R. W." } }, { "id": "Saxl-J", "name": { "family": "Saxl", "given": "J." } } ] }, "doi": "10.1007/978-94-011-5308-9_18", "resource_type": "book_section", "pub_year": "1998", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/gp5x2-23y78", "eprint_id": 544, "eprint_status": "archive", "datestamp": "2023-08-22 11:31:24", "lastmod": "2023-10-13 21:51:59", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Finite groups acting on homology manifolds", "ispublished": "pub", "full_text_status": "public", "note": "Pacific journal of mathematics, Vol. 181, No. 3, 1997 - Dedicated to the Memory of Olga Taussky-Todd. \n\nThis work was partially supported by NSF DMS-9101237 and NSF DMS-9622843.", "abstract": "In this paper we study homology manifolds T admitting the action of a finite group preserving the structure of a regular CW-complex on T. The CW-complex is parameterized by a poset and the topological properties of the manifold are translated into a combinatorial setting via the poset. We concentrate on n-manifolds which admit a fairly rigid group of automorphisms transitive on the n-cells of the complex. This allows us to make yet another translation from a combinatorial into a group theoretic setting. We close by using our machinery to construct representations on manifolds of the Monster, the largest sporadic group. Some of these manifolds are of dimension 24, and hence candidates for examples to Hirzebruch's Prize Question in [HBJ], but unfortunately closer inspection shows the A^-genus of these manifolds is 0 rather than 1, so none is a Hirzebruch manifold.", "date": "1997-01-01", "date_type": "published", "publication": "Pacific Journal of Mathematics", "volume": "181", "number": "3", "publisher": "Pacific Journal of Mathematics", "pagerange": "3-36", "id_number": "CaltechAUTHORS:ASCpjm97", "issn": "0030-8730", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:ASCpjm97", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "primary_object": { "basename": "ASCpjm97.pdf", "url": "https://authors.library.caltech.edu/records/gp5x2-23y78/files/ASCpjm97.pdf" }, "resource_type": "article", "pub_year": "1997", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/x9yyz-sa445", "eprint_id": 103145, "eprint_status": "archive", "datestamp": "2023-08-22 09:24:54", "lastmod": "2023-10-20 15:41:59", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Simple connectivity of p-group complexes", "ispublished": "pub", "full_text_status": "restricted", "keywords": "Finite Group; Simple Group; Prime Divisor; Homotopy Type; Outer Automorphism", "note": "\u00a9 1993 Springer Verlag. \n\nReceived 22 June 1992; Revised 07 September 1992; Issue Date June 1993. \n\nTo John Thompson on the occasion of his receipt of the Wolf Prize. \n\nThis work was partially supported by NSF DMS-8721480 and NSA MDA90-88-H-2032.", "abstract": "We investigate the simple connectivity ofp-subgroup complexes of finite groups.", "date": "1993-06", "date_type": "published", "publication": "Israel Journal of Mathematics", "volume": "82", "number": "1-3", "publisher": "Springer", "pagerange": "1-43", "id_number": "CaltechAUTHORS:20200512-131404038", "issn": "0021-2172", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200512-131404038", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-8721480" }, { "agency": "National Security Agency", "grant_number": "MDA90-88-H-2032" } ] }, "doi": "10.1007/bf02808107", "resource_type": "article", "pub_year": "1993", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/0xgqd-p8x79", "eprint_id": 103143, "eprint_status": "archive", "datestamp": "2023-08-22 09:01:29", "lastmod": "2023-10-20 15:41:44", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "M." } }, { "id": "Segev-Y", "name": { "family": "Segev", "given": "Y." } } ] }, "title": "Locally connected simplicial maps", "ispublished": "pub", "full_text_status": "restricted", "keywords": "Fundamental Group; Local System; Simplicial Complex; Order Complex; Connect SIMPLICIAL", "note": "\u00a9 1992 Springer-Verlag. \n\nReceived 05 May 1991; Revised 11 November 1991; Issue Date October 1992. \n\nThis work was partially supported by BSF 88-00164. The first author is partially supported by NSF DMS-8721480 and NSA MDA90-88-H-2032.", "abstract": "In Propositions 1.6 and 7.6 of his paper onp-group complexes of finite groups [5], Quillen establishes fundamental results comparing the homology and the fundamental group of the order complexes of posetsP, Q admitting a mapf :P \u2192Q of posets with good local behavior. We prove the analogue of Quillen's results for mapsf :K\u2192L of simplicial complexesK andL in a more general setup.", "date": "1992-10", "date_type": "published", "publication": "Israel Journal of Mathematics", "volume": "77", "number": "3", "publisher": "Springer", "pagerange": "285-303", "id_number": "CaltechAUTHORS:20200512-125900851", "issn": "0021-2172", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200512-125900851", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "88-00164" }, { "agency": "NSF", "grant_number": "DMS-8721480" }, { "agency": "National Security Agency", "grant_number": "MDA90-88-H-2032" } ] }, "doi": "10.1007/bf02773693", "resource_type": "article", "pub_year": "1992", "author_list": "Aschbacher, M. and Segev, Y." }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/9835e-b1834", "eprint_id": 103120, "eprint_status": "archive", "datestamp": "2023-08-22 08:33:21", "lastmod": "2023-10-20 15:40:10", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Kleidman-P-B", "name": { "family": "Kleidman", "given": "Peter B." } }, { "id": "Liebeck-M-W", "name": { "family": "Liebeck", "given": "Martin W." } } ] }, "title": "Exponents of almost simple groups and an application to the restricted Burnside problem", "ispublished": "pub", "full_text_status": "restricted", "keywords": "Simple Group; Outer Automorphism; Finite Simple Group; Split Extension; Outer Automorphism Group", "note": "\u00a9 1991 Springer-Verlag. \n\nReceived 04 August 1989; Accepted 22 November 1990; Issue Date December 1991. \n\nPartially supported by NSF DMS-8721480 and NSA MDA 90-88-H-2032.", "abstract": "This paper is motivated by: The Restricted Burnside Problem R(n). For each r, are there only finitely many r-generator finite groups of exponent n?", "date": "1991-12", "date_type": "published", "publication": "Mathematische Zeitschrift", "volume": "208", "number": "1", "publisher": "Springer Verlag", "pagerange": "401-409", "id_number": "CaltechAUTHORS:20200512-074240672", "issn": "0025-5874", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200512-074240672", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-8721480" }, { "agency": "National Security Agency", "grant_number": "MDA 90-88-H-2032" } ] }, "doi": "10.1007/bf02571536", "resource_type": "article", "pub_year": "1991", "author_list": "Aschbacher, Michael; Kleidman, Peter B.; et el." }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/pbths-spv27", "eprint_id": 103122, "eprint_status": "archive", "datestamp": "2023-08-20 00:39:09", "lastmod": "2023-10-20 15:40:24", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Segev-Y", "name": { "family": "Segev", "given": "Yoav" } } ] }, "title": "The uniqueness of groups of type J\u2084", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 1991 Springer-Verlag. \n\nOblatum VIII-1990 & 31-I-1991. \n\nThis work was partially supported by BSF 88-00164. The first author is partially supported by NSF DMS-8721480 and NSA MDA 90-88-H-2032.", "abstract": "We give the first computer free proof of the uniqueness of groups of type J\u2084. In addition we supply simplified proofs of some properties of such groups, such as the structure of certain subgroups.\nA group of type J\u2084 is a finite group G possessing an involution z such that H=C_G(z) satisfies F*(H)=Q is extraspecial of order 2\u00b9\u00b3, H/Q is isomorphic to Z\u2083 extended by Aut (M\u2082\u2082), and z^G \u22c2 Q \u2260 {z}. We prove: Main Theorem. Up to isomorphism there exists at most one group of type J\u2084.", "date": "1991-12", "date_type": "published", "publication": "Inventiones Mathematicae", "volume": "105", "number": "1", "publisher": "Springer", "pagerange": "589-607", "id_number": "CaltechAUTHORS:20200512-075747975", "issn": "0020-9910", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200512-075747975", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "Binational Science Foundation (USA-Israel)", "grant_number": "88-00164" }, { "agency": "NSF", "grant_number": "DMS-8721480" }, { "agency": "National Security Agency", "grant_number": "MDA 90-88-H-2032" } ] }, "doi": "10.1007/bf01232280", "resource_type": "article", "pub_year": "1991", "author_list": "Aschbacher, Michael and Segev, Yoav" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/tzbk6-99d36", "eprint_id": 80053, "eprint_status": "archive", "datestamp": "2023-08-19 23:21:13", "lastmod": "2023-10-17 15:28:01", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "On conjectures of Guralnick and Thompson", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 1990 Academic Press, Inc. \n\nReceived 9 June 1988. \n\nPartially supported by NSF Grant DMS-8721480 and NSA Grant MDA 90-88-H-2032.", "abstract": "Given a permutation s on a finite set \u03a9 of order n, define c(s) to be the number of cycles of sand Ind(s) = n - c(s).\nDefine a genus g system to be a triple ( G, \u03a9, S), where \u03a9 is a finite set, G is a transitive subgroup of Sym(\u03a9), and S = (g_j: 1 \u2a7dj\u2a7dr is a family of elements of G^# such that G = \u27e8S\u27e9, g_1...g_r = 1, and 2(\u2758\u03a9\u2758 + g-1)= \u2211_(j=1) Ind(g_j).", "date": "1990-12", "date_type": "published", "publication": "Journal of Algebra", "volume": "135", "number": "2", "publisher": "Elsevier", "pagerange": "277-343", "id_number": "CaltechAUTHORS:20170810-072647371", "issn": "0021-8693", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170810-072647371", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-8721480" }, { "agency": "NSF", "grant_number": "MDA 90-88-H-2032" } ] }, "doi": "10.1016/0021-8693(90)90292-V", "resource_type": "article", "pub_year": "1990", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/dpjep-vcf30", "eprint_id": 103119, "eprint_status": "archive", "datestamp": "2023-08-19 23:04:37", "lastmod": "2023-10-20 15:40:07", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } }, { "id": "Kleidman-P-B", "name": { "family": "Kleidman", "given": "Peter B." } } ] }, "title": "On a conjecture of Quillen and a lemma of Robinson", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 1990 Birkh\u00e4user Verlag, Basel. \n\nReceived 20 February 1989; Issue Date September 1990.", "abstract": "In this note we are concerned with finite groups G and primes p satisfying the Robinson Properties.", "date": "1990-09", "date_type": "published", "publication": "Archiv der Mathematik", "volume": "55", "number": "3", "publisher": "Springer", "pagerange": "209-217", "id_number": "CaltechAUTHORS:20200512-073808676", "issn": "0003-889X", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200512-073808676", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.1007/bf01191159", "resource_type": "article", "pub_year": "1990", "author_list": "Aschbacher, Michael and Kleidman, Peter B." }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/pgbqn-knm13", "eprint_id": 103116, "eprint_status": "archive", "datestamp": "2023-08-22 07:44:04", "lastmod": "2023-10-20 15:39:56", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "The existence of J\u2083 and its embeddings in E\u2086", "ispublished": "pub", "full_text_status": "restricted", "keywords": "Chevalley Group; Existence Proof; Sporadic Group; Previous Existence; Simple Chevalley Group", "note": "\u00a9 1990 Kluwer Academic Publishers. \n\nReceived 25 August 1989; Issue Date September 1990. \n\nTo Jacques Tits on his sixtieth birthday. \n\nPartially supported by NSF DMS-8721480 and NSA MDA90-88-H-2032.", "abstract": "We determine the embeddings of the third sporadic group J\u2083 of Janko in simple Chevalley groups of type E\u2086 over finite and algebraically closed fields. As a corollary we obtain a short elegant existence proof of J\u2083. This is of interest as J\u2083 is one of the few sporadic groups not contained in the Monster, so its existence cannot be verified within that group. Previous existence proofs were highly computational; cf. [4] and [6].", "date": "1990-09", "date_type": "published", "publication": "Geometriae Dedicata", "volume": "35", "number": "1-3", "publisher": "Springer", "pagerange": "143-154", "id_number": "CaltechAUTHORS:20200512-071755284", "issn": "0046-5755", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200512-071755284", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF", "grant_number": "DMS-8721480" }, { "agency": "National Security Agency", "grant_number": "MDA 90-88-H-2032" } ] }, "doi": "10.1007/bf00147344", "resource_type": "article", "pub_year": "1990", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/sfx6b-9r313", "eprint_id": 105683, "eprint_status": "archive", "datestamp": "2023-08-19 20:19:25", "lastmod": "2023-10-20 22:17:52", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Some multilinear forms with large isometry groups", "ispublished": "pub", "full_text_status": "restricted", "note": "\u00a9 1988 by D. Reidel Publishing Company. \n\nPartially supported by the National Science Foundation.", "abstract": "Many groups are best described as the group of automorphisms of some natural object. I'm interested in obtaining such descriptions of the finite simple groups, and more generally descriptions of the groups of Lie type over arbitrary fields. The representation of the alternating group of degree n as the group of automorphisms of a set of order n\nis an excellent example of such a description. The representation of the classical groups as the isometry groups of bilinear or sequilinear forms is another.", "date": "1988-01", "date_type": "published", "publication": "Geometriae Dedicata", "volume": "25", "number": "1-3", "publisher": "Springer", "pagerange": "417-465", "id_number": "CaltechAUTHORS:20200930-113055206", "issn": "0046-5755", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200930-113055206", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF" } ] }, "doi": "10.1007/bf00191936", "resource_type": "article", "pub_year": "1988", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/efe7x-yvd92", "eprint_id": 105682, "eprint_status": "archive", "datestamp": "2023-08-19 20:11:40", "lastmod": "2023-10-20 22:17:46", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Some Multilinear Forms with Large Isometry Groups", "ispublished": "unpub", "full_text_status": "restricted", "note": "\u00a9 D. Reidel Publishing Company 1988. \n\nPartially supported by the National Science Foundation.", "abstract": "Many groups are best described as the group of automorphisms of some natural object. I'm interested in obtaining such descriptions of the finite simple groups, and more generally descriptions of the groups of Lie type over arbitrary fields. The representation of the alternating group of degree n as the group of automorphisms of a set of order n is an excellent example of such a description. The representation of the classical groups as the isometry groups of bilinear or sequilinear forms is another.", "date": "1988", "date_type": "published", "publisher": "Springer Netherlands", "place_of_pub": "Dordrecht", "pagerange": "417-465", "id_number": "CaltechAUTHORS:20200930-113055093", "isbn": "9789401082822", "book_title": "Geometries and Groups", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200930-113055093", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF" } ] }, "contributors": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "M." } }, { "id": "Cohen-A-M", "name": { "family": "Cohen", "given": "A. M." } }, { "id": "Kantor-W-M", "name": { "family": "Kantor", "given": "W. M." } } ] }, "doi": "10.1007/978-94-009-4017-8_15", "resource_type": "book_section", "pub_year": "1988", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/my4ef-an845", "eprint_id": 97883, "eprint_status": "archive", "datestamp": "2023-08-22 05:08:11", "lastmod": "2023-10-18 16:50:38", "type": "book", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "M." } }, { "id": "Cohen-A-M", "name": { "family": "Cohen", "given": "A. M." } }, { "id": "Kantor-W-M", "name": { "family": "Kantor", "given": "W. M." } } ] }, "title": "Geometries and Groups", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Area; Finite; Lie; Maxima; Morphism; Node.js; algebra; character; classification; diagrams; presentation; reflection; representation theory; set; techniques", "note": "\u00a9 1988 Springer Science+Business Media B.V.", "abstract": "The workshop was set up in order to stimulate the interaction between (finite and algebraic) geometries and groups. Five areas of concentrated research were chosen on which attention would be focused, namely: diagram geometries and chamber systems with transitive automorphism groups, geometries viewed as incidence systems, properties of finite groups of Lie type, geometries related to finite simple groups, and algebraic groups. The list of talks (cf. page iii) illustrates how these subjects were represented during the workshop. The contributions to these proceedings mainly belong to the first three areas; therefore, (i) diagram geometries and chamber systems with transitive automorphism groups, (ii) geometries viewed as incidence systems, and (iii) properties of finite groups of Lie type occur as section titles. The fourth and final section of these proceedings has been named graphs and groups; besides some graph theory, this encapsules most of the work related to finite simple groups that does not (explicitly) deal with diagram geometry. A few more words about the content: (i). Diagram geometries and chamber systems with transitive automorphism groups. As a consequence of Tits' seminal work on the subject, all finite buildings are known. But usually, in a situation where groups are to be characterized by certain data concerning subgroups, a lot less is known than the full parabolic picture corresponding to the building.", "date": "1987", "date_type": "published", "publisher": "Springer", "place_of_pub": "Dordrecht", "id_number": "CaltechAUTHORS:20190814-100149802", "isbn": "978-94-010-8282-2", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190814-100149802", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "contributors": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "M." } }, { "id": "Cohen-A-M", "name": { "family": "Cohen", "given": "A. M." } }, { "id": "Kantor-W-M", "name": { "family": "Kantor", "given": "W. M." } } ] }, "doi": "10.1007/978-94-009-4017-8", "resource_type": "book", "pub_year": "1987", "author_list": "Aschbacher, M.; Cohen, A. M.; et el." }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/rddsk-b2514", "eprint_id": 105732, "eprint_status": "archive", "datestamp": "2023-08-22 04:38:34", "lastmod": "2024-01-15 17:05:57", "type": "book_section", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Finite simple groups and their subgroups", "ispublished": "unpub", "full_text_status": "restricted", "keywords": "Finite Group; Simple Group; Maximal Subgroup; Finite Simple Group; Permutation Representation", "note": "\u00a9 Springer-Verlag 1986.\n\nThe author would like to thank the organizers of the Peking Symposium, particularly Professor Hsio-Fu Tuan, for all their efforts which made the conference and this article possible. \n\nThe author's work is partially supported by the National Science Foundation.", "abstract": "The material in this article corresponds roughly to the contents of six lectures given at the International Symposium on Group Theory at Peking University in September 1984. \n\nIn essence the article describes the beginnings of a theory of permutation representations of finite groups based on the classifications of the finite simple groups. Chapter 3 is devoted to an outline of the Classification, with emphasis on recent efforts to improve the proof of the Classification Theorem. Chapters, 1, 2, and 6 discuss finite groups themselves, a notion of geometry due to J. Tits, and a class of group theoretical techniques introduced by B. Fischer. Each of these topics plays a role in the theory of permutation representations under discussion. The heart of the theory is the study of the subgroup structure of the finite simple groups. Certain results on this structure are described in Chapter 4 and 5.", "date": "1986", "date_type": "published", "publisher": "Springer", "place_of_pub": "Berlin, Heidelberg", "pagerange": "1-57", "id_number": "CaltechAUTHORS:20201001-145810955", "isbn": "9783540164562", "book_title": "Group Theory, Beijing 1984", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20201001-145810955", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "funders": { "items": [ { "agency": "NSF" } ] }, "contributors": { "items": [ { "id": "Hsio-Fu-Tuan", "name": { "family": "Hsio-Fu", "given": "Tuan" } } ] }, "doi": "10.1007/bfb0076170", "resource_type": "book_section", "pub_year": "1986", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/bh4sn-ayp61", "eprint_id": 97884, "eprint_status": "archive", "datestamp": "2023-08-22 02:22:13", "lastmod": "2023-10-18 16:50:43", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "Classification of the Finite Simple Groups", "ispublished": "pub", "full_text_status": "public", "keywords": "Finite Group; Simple Group; Prime Order; Component Type; Chevalley Group", "note": "\u00a9 1980 Springer Science+Business Media, Inc.", "abstract": "The classification of the finite simple groups was completed sometime during the summer of 1980. To the extent that I can reconstruct things, the last piece in the puzzle was filled in by Ronald Solomon of Ohio State University. At the other chronological extreme, the theory of finite groups can be traced back to its beginnings in the early nineteenth century in the work of Abel, Cauchy, and Galois. Hence the problem of classifying the finite simple groups has a history of over a century and a half. The proof of the Classification Theorem is made up of thousands of pages in various mathematical journals with at least another thousand pages still left to appear in print. Many mathematicians have contributed to the proof; some have spent their entire mathematical lives working\non the problem.", "date": "1980-06", "date_type": "published", "publication": "Mathematical Intelligencer", "volume": "3", "number": "2", "publisher": "Springer", "pagerange": "59-65", "id_number": "CaltechAUTHORS:20190814-100802612", "issn": "0343-6993", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190814-100802612", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "doi": "10.1007/bf03022850", "resource_type": "article", "pub_year": "1980", "author_list": "Aschbacher, Michael" }, { "id": "https://authors.library.caltech.eduhttps://authors.library.caltech.edu/records/679ps-2b104", "eprint_id": 542, "eprint_status": "archive", "datestamp": "2023-08-22 00:18:41", "lastmod": "2023-10-13 21:51:55", "type": "article", "metadata_visibility": "show", "creators": { "items": [ { "id": "Aschbacher-M", "name": { "family": "Aschbacher", "given": "Michael" } } ] }, "title": "A characterization of the unitary and symplectic groups over finite fields of characteristic at least $5$", "ispublished": "pub", "full_text_status": "public", "note": "Received February 25, 1972 and in revised form February 26, 1973. \n\nEuclid Identifier: euclid.pjm/1102946071 \nZentralblatt Math Identifier : 0299.20009 \nMathmatical Reviews number (MathSciNet): MR0338151", "abstract": "The following characterization is obtained: \n\nTHEOREM. Let G be a finite group generated by a conjugacy class D of subgroups of prime order p ^ 5, such that for any choice of distinct A and B in D, the subgroup generated by A and B is isomorphic to Zp x Zp, L2(pm) or SL2(pm), where m depends on A and B. Assume G has no nontrivial solvable normal subgroup. Then G is isomorphic to Spn(q) or Un(q) for some power q of p.", "date": "1973", "date_type": "published", "publication": "Pacific Journal of Mathematics", "volume": "47", "number": "1", "publisher": "Pacific Journal of Mathematics", "pagerange": "5-26", "id_number": "CaltechAUTHORS:ASCpjm73", "issn": "0030-8730", "official_url": "https://resolver.caltech.edu/CaltechAUTHORS:ASCpjm73", "rights": "No commercial reproduction, distribution, display or performance rights in this work are provided.", "primary_object": { "basename": "ASCpjm73.pdf", "url": "https://authors.library.caltech.edu/records/679ps-2b104/files/ASCpjm73.pdf" }, "resource_type": "article", "pub_year": "1973", "author_list": "Aschbacher, Michael" } ]