{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/19169"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/19169","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Blocks and virtually irreducible lattices","abstract":"Theorem (5.11). Let H be a subgroup of G and let b be an admissible block of SH with defect group D. If every automorphism of D which preserves conjugacy classes is an inner automorphism, then there is a virtually irreducible SH-lattice in b with vertex D such that U$\\sp{\\rm G}$ = V $\\oplus$ W with V virtually irreducible and U $\\not\\vert$ W$\\sb{\\rm H}$.","abstract_html":"Theorem (5.11). Let H be a subgroup of G and let b be an admissible block of SH with defect group D. If every automorphism of D which preserves conjugacy classes is an inner automorphism, then there is a virtually irreducible SH-lattice in b with vertex D such that U$\\sp{\\rm G}$ = V $\\oplus$ W with V virtually irreducible and U $\\not\\vert$ W$\\sb{\\rm H}$.","abstract_has_math":true,"creators":["Ellers, Harald Erich Herbert"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Dade, Everett C."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T11:59:03Z","date_published":"2011-05-07T11:59:03Z","updated_at":"2026-07-22T22:25:12Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1989 Ellers, Harald Erich Herbert"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9010852","(UMI)AAI9010852"],"render_values":[{"text":"AAI9010852","href":null,"code":true},{"text":"(UMI)AAI9010852","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/19169","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Dade, Everett C."]},{"key":"dc:creator","label":"Author","values":["Ellers, Harald Erich Herbert"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T11:59:03Z","10000-01-01","1989"]},{"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 1989 Ellers, Harald Erich Herbert"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9010852","(UMI)AAI9010852","http://hdl.handle.net/2142/19169"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Theorem (5.11). Let H be a subgroup of G and let b be an admissible block of SH with defect group D. If every automorphism of D which preserves conjugacy classes is an inner automorphism, then there is a virtually irreducible SH-lattice in b with vertex D such that U$\\sp{\\rm G}$ = V $\\oplus$ W with V virtually irreducible and U $\\not\\vert$ W$\\sb{\\rm H}$.","We use R. Knorr's theory of virtually irreducible lattices to study the blocks of a finite group.","Let G be a finite group and let p be a rational prime. Let R be a complete discrete valuation ring of characteristic zero with maximal ideal generated by $\\pi$ and with p $\\varepsilon$ $\\pi$R. Let K be the field of fractions of R, and let R = R/$\\pi$R. Assume that R is algebraically closed and that K is a splitting field for every subgroup of G.","Knorr showed that any indecomposable RG-lattice of height zero is virtually irreducible. We use this fact to generalize Brauer's Third Main Theorem on Blocks as follows.","Theorem (3.1). Let B be a block of RG, and let M be an indecomposable RG-lattice in B of height zero. Suppose that H is a subgroup of G and that b is an admissible block of RH. Then b$\\sp{\\rm G}$ = B if and only if b contains an indecomposable component of M$\\sb{\\rm H}$ of height zero.","We also prove the following connection between Brauer correspondence of blocks and induction of virtually irreducible lattices.","Theorem (5.2). Let H be a subgroup of G and let b be a block of RH. Suppose that there is a virtually irreducible RH-lattice U in b such that U$\\sp{\\rm G}$ = V $\\oplus$ W with V virtually irreducible and U $\\not\\vert$ W$\\sb{\\rm H}$. Then b$\\sp{\\rm G}$ is defined and V is in b$\\sp{\\rm G}$.","Most admissible blocks contain a virtually irreducible lattice U as in Theorem (5.2); there is a finite extension S of R such that the following is true.","We also investigate the question: if B is a block of RG and if there is a block pair (D,b) in G with b$\\sp{\\rm G}$ = B, is there a virtually irreducible RG-lattice in B with vertex D? Theorem (5.11) gives a sufficient condition on D for this question to have an affirmative answer, provided we replace R by a certain finite extension. We give several more conditions of this kind. This is a partial converse to a theorem of Knorr.","Made available in DSpace on 2011-05-07T11:59:03Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9010852.pdf: 1938741 bytes, checksum: 6cca28f2afef2fb71a968005ac88b311 (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:35:07Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:13:42-05:00 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":["Blocks and virtually irreducible lattices"]}]}],"canonical_facts":{"dc:contributor":["Dade, Everett C."],"dc:creator":["Ellers, Harald Erich Herbert"],"dc:date":["2011-05-07T11:59:03Z","10000-01-01","1989"],"dc:description":["Theorem (5.11). Let H be a subgroup of G and let b be an admissible block of SH with defect group D. If every automorphism of D which preserves conjugacy classes is an inner automorphism, then there is a virtually irreducible SH-lattice in b with vertex D such that U$\\sp{\\rm G}$ = V $\\oplus$ W with V virtually irreducible and U $\\not\\vert$ W$\\sb{\\rm H}$.","We use R. Knorr's theory of virtually irreducible lattices to study the blocks of a finite group.","Let G be a finite group and let p be a rational prime. Let R be a complete discrete valuation ring of characteristic zero with maximal ideal generated by $\\pi$ and with p $\\varepsilon$ $\\pi$R. Let K be the field of fractions of R, and let R = R/$\\pi$R. Assume that R is algebraically closed and that K is a splitting field for every subgroup of G.","Knorr showed that any indecomposable RG-lattice of height zero is virtually irreducible. We use this fact to generalize Brauer's Third Main Theorem on Blocks as follows.","Theorem (3.1). Let B be a block of RG, and let M be an indecomposable RG-lattice in B of height zero. Suppose that H is a subgroup of G and that b is an admissible block of RH. Then b$\\sp{\\rm G}$ = B if and only if b contains an indecomposable component of M$\\sb{\\rm H}$ of height zero.","We also prove the following connection between Brauer correspondence of blocks and induction of virtually irreducible lattices.","Theorem (5.2). Let H be a subgroup of G and let b be a block of RH. Suppose that there is a virtually irreducible RH-lattice U in b such that U$\\sp{\\rm G}$ = V $\\oplus$ W with V virtually irreducible and U $\\not\\vert$ W$\\sb{\\rm H}$. Then b$\\sp{\\rm G}$ is defined and V is in b$\\sp{\\rm G}$.","Most admissible blocks contain a virtually irreducible lattice U as in Theorem (5.2); there is a finite extension S of R such that the following is true.","We also investigate the question: if B is a block of RG and if there is a block pair (D,b) in G with b$\\sp{\\rm G}$ = B, is there a virtually irreducible RG-lattice in B with vertex D? Theorem (5.11) gives a sufficient condition on D for this question to have an affirmative answer, provided we replace R by a certain finite extension. We give several more conditions of this kind. This is a partial converse to a theorem of Knorr.","Made available in DSpace on 2011-05-07T11:59:03Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9010852.pdf: 1938741 bytes, checksum: 6cca28f2afef2fb71a968005ac88b311 (MD5) Previous issue date: 1989","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:35:07Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:13:42-05:00 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":["AAI9010852","(UMI)AAI9010852","http://hdl.handle.net/2142/19169"],"dc:language":["eng"],"dc:rights":["Copyright 1989 Ellers, Harald Erich Herbert"],"dc:subject":["Mathematics"],"dc:title":["Blocks and virtually irreducible lattices"],"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":"2026-07-22T22:25:12Z"}