{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/19660"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/19660","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Verification of the McKay-Alperin-Dade Conjecture for the covering groups of the Mathieu group M(22)","abstract":"The McKay-Alperin-Dade Conjecture states that the number of complex irreducible characters with a given defect d in a p-block B of a finite group G can be expressed in terms of an alternating sum of the numbers of complex irreducible characters with related defects $d\\sp\\prime$ in related p-blocks $B\\sp\\prime$ of the normalizers $N\\sb{G}(C)$ of representatives C of the G-conjugacy classes of radical p-chains of G. Specifically, we have the following.","abstract_html":"The McKay-Alperin-Dade Conjecture states that the number of complex irreducible characters with a given defect d in a p-block B of a finite group G can be expressed in terms of an alternating sum of the numbers of complex irreducible characters with related defects $d\\sp\\prime$ in related p-blocks $B\\sp\\prime$ of the normalizers $N\\sb{G}(C)$ of representatives C of the G-conjugacy classes of radical p-chains of G. Specifically, we have the following.","abstract_has_math":true,"creators":["Huang, Margaret Janice Fernald"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Suzuki, Michio"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:14:28Z","date_published":"2011-05-07T12:14:28Z","updated_at":"2026-07-22T22:25:14Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1992 Huang, Margaret Janice Fernald"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9305561","(UMI)AAI9305561"],"render_values":[{"text":"AAI9305561","href":null,"code":true},{"text":"(UMI)AAI9305561","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/19660","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Suzuki, Michio"]},{"key":"dc:creator","label":"Author","values":["Huang, Margaret Janice Fernald"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:14:28Z","10000-01-01","1992"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1992 Huang, Margaret Janice Fernald"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9305561","(UMI)AAI9305561","http://hdl.handle.net/2142/19660"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["The McKay-Alperin-Dade Conjecture states that the number of complex irreducible characters with a given defect d in a p-block B of a finite group G can be expressed in terms of an alternating sum of the numbers of complex irreducible characters with related defects $d\\sp\\prime$ in related p-blocks $B\\sp\\prime$ of the normalizers $N\\sb{G}(C)$ of representatives C of the G-conjugacy classes of radical p-chains of G. Specifically, we have the following.","Conjecture A (The McKay-Alperin-Dade conjecture). If $O\\sb{p}(G)$ is the Sylow p-subgroup of a central subgroup N of G, and is not a defect group of B then $$\\sum\\limits\\sb{C\\in{\\cal R}/G}(-1)\\sp{\\vert C\\vert}k(N\\sb{G}(C),B,d,O\\vert\\nu) = 0$$for any $O\\le Out(G\\vert N),$ where $\\nu$ is a linear character of N and ${\\cal R}/G$ is our family of representatives.","This paper presents a verification of the M-A-D Conjecture for the group 12.$M\\sb{22},$ whose order is $2\\sp9\\cdot3\\sp3\\cdot5\\cdot7\\cdot11.$ Since Dade has shown that Conjecture A holds for any blocks with cyclic defect groups, this paper deals specifically with the primes 3 and 2. In each case, representatives of the $M\\sb{22}$-conjugacy classes of the radical p-subgroups of $M\\sb{22}$ are identified, together with their normalizers. Subsequently, a complete listing of the representatives of the $M\\sb{22}$-conjugacy classes of radical p-chains C, together with their normalizers $N\\sb{M\\sb{22}}(C)$ is made.","The normalizers $N\\sb{n.M\\sb{22}}(C)$ are then determined for each radical p-chain C and for n = 1,2,3,4,6,12. The action of the outer automorphism group of $M\\sb{22},$ which is cyclic of order 2, on each of the groups $N\\sb{n.M\\sb{22}}(C)$ is identified. Finally, the M-A-D Conjecture is verified for the 3-blocks and the 2-blocks of $n.M\\sb{22}.$","Made available in DSpace on 2011-05-07T12:14:28Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9305561.pdf: 7153690 bytes, checksum: b511e0831c7f74a3cbf15c4a854385dc (MD5) Previous issue date: 1992","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:38:34Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:16:05-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Verification of the McKay-Alperin-Dade Conjecture for the covering groups of the Mathieu group M(22)"]}]}],"canonical_facts":{"dc:contributor":["Suzuki, Michio"],"dc:creator":["Huang, Margaret Janice Fernald"],"dc:date":["2011-05-07T12:14:28Z","10000-01-01","1992"],"dc:description":["The McKay-Alperin-Dade Conjecture states that the number of complex irreducible characters with a given defect d in a p-block B of a finite group G can be expressed in terms of an alternating sum of the numbers of complex irreducible characters with related defects $d\\sp\\prime$ in related p-blocks $B\\sp\\prime$ of the normalizers $N\\sb{G}(C)$ of representatives C of the G-conjugacy classes of radical p-chains of G. Specifically, we have the following.","Conjecture A (The McKay-Alperin-Dade conjecture). If $O\\sb{p}(G)$ is the Sylow p-subgroup of a central subgroup N of G, and is not a defect group of B then $$\\sum\\limits\\sb{C\\in{\\cal R}/G}(-1)\\sp{\\vert C\\vert}k(N\\sb{G}(C),B,d,O\\vert\\nu) = 0$$for any $O\\le Out(G\\vert N),$ where $\\nu$ is a linear character of N and ${\\cal R}/G$ is our family of representatives.","This paper presents a verification of the M-A-D Conjecture for the group 12.$M\\sb{22},$ whose order is $2\\sp9\\cdot3\\sp3\\cdot5\\cdot7\\cdot11.$ Since Dade has shown that Conjecture A holds for any blocks with cyclic defect groups, this paper deals specifically with the primes 3 and 2. In each case, representatives of the $M\\sb{22}$-conjugacy classes of the radical p-subgroups of $M\\sb{22}$ are identified, together with their normalizers. Subsequently, a complete listing of the representatives of the $M\\sb{22}$-conjugacy classes of radical p-chains C, together with their normalizers $N\\sb{M\\sb{22}}(C)$ is made.","The normalizers $N\\sb{n.M\\sb{22}}(C)$ are then determined for each radical p-chain C and for n = 1,2,3,4,6,12. The action of the outer automorphism group of $M\\sb{22},$ which is cyclic of order 2, on each of the groups $N\\sb{n.M\\sb{22}}(C)$ is identified. Finally, the M-A-D Conjecture is verified for the 3-blocks and the 2-blocks of $n.M\\sb{22}.$","Made available in DSpace on 2011-05-07T12:14:28Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9305561.pdf: 7153690 bytes, checksum: b511e0831c7f74a3cbf15c4a854385dc (MD5) Previous issue date: 1992","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:38:34Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:16:05-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9305561","(UMI)AAI9305561","http://hdl.handle.net/2142/19660"],"dc:language":["eng"],"dc:rights":["Copyright 1992 Huang, Margaret Janice Fernald"],"dc:subject":["Mathematics"],"dc:title":["Verification of the McKay-Alperin-Dade Conjecture for the covering groups of the Mathieu group M(22)"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:14Z"}