{"id":{"repo_id":"birmingham","oai_identifier":"oai:etheses.bham.ac.uk:86"},"canonical_url":"https://search.dev.ndltd.org/etd/birmingham/oai:etheses.bham.ac.uk:86","repository":{"repo_id":"birmingham","name":"University of Birmingham","base_url":"https://etheses.bham.ac.uk/cgi/oai2"},"display":{"title":"The Ordinary Weight conjecture and Dade's Projective Conjecture for p-blocks with an extra-special defect group","abstract":"Let \\(p\\) be a rational odd prime number, \\(G\\) be a finite group such that \\(|G|=p^am\\), with \\(p \\nmid m\\). Let \\(B\\) be a \\(p\\)-block of \\(G\\) with a defect group \\(E\\) which is an extra-special \\(p\\)-group of order \\(p^3\\) and exponent \\(p\\). Consider a fixed maximal \\((G, B)\\)-subpair \\((E, b_E)\\). Let \\(b\\) be the Brauer correspondent of \\(B\\) for \\(N_G(E, b_E)\\). For a non-negative integer \\(d\\), let \\(k_d(B)\\) denote the number of irreducible characters \\(\\chi\\) in \\(B\\) which have \\(\\chi(1)_p=p^{a-d}\\) and let \\(k_d(b)\\) be the corresponding number of \\(b\\). Various generalizations of Alperin's Weight Conjecture and McKay's Conjecture are due to Reinhard Knorr, Geoffrey R. Robinson and Everett C. Dade. We follow Geoffrey R. Robinson's approach to consider the Ordinary Weight Conjecture, and Dade's Projective Conjecture. The general question is whether it follows from either of the latter two conjectures that \\(k_d(B)=k_d(b)\\) for all \\(d\\) for the \\(p\\)-block \\(B\\). The objective of this thesis is to show that these conjectures predict that \\(k_d(B)=k_d(b)\\), for all non-negative integers \\(d\\). It is well known that \\(N_G(E, b_E)/EC_G(E)\\) is a \\(p^'\\)-subgroup of the automorphism group of \\(E\\). Hence, we have considered some special cases of the above question.The unique largest normal \\(p\\)-subgroup of \\(G\\), \\(O_p(G)\\) is the central focus of our attention. We consider the case that \\(O_p(G)\\) is a central \\(p\\)-subgroup of \\(G\\), as well as the case that \\(O_p(G)\\) is not central. In both cases, the common factor is that \\(O_p(G)\\) is strictly contained in the defect group of \\(B\\).","abstract_html":"Let \\(p\\) be a rational odd prime number, \\(G\\) be a finite group such that <span class=\"etd-inline-math\">|G|=p<sup>a</sup>m</span>, with \\(p \\nmid m\\). Let \\(B\\) be a \\(p\\)-block of \\(G\\) with a defect group \\(E\\) which is an extra-special \\(p\\)-group of order <span class=\"etd-inline-math\">p<sup>3</sup></span> and exponent \\(p\\). Consider a fixed maximal \\((G, B)\\)-subpair <span class=\"etd-inline-math\">(E, b<sub>E</sub>)</span>. Let \\(b\\) be the Brauer correspondent of \\(B\\) for <span class=\"etd-inline-math\">N<sub>G</sub>(E, b<sub>E</sub>)</span>. For a non-negative integer \\(d\\), let <span class=\"etd-inline-math\">k<sub>d</sub>(B)</span> denote the number of irreducible characters \\(\\chi\\) in \\(B\\) which have <span class=\"etd-inline-math\">\\chi(1)<sub>p</sub>=p<sup>a-d</sup></span> and let <span class=\"etd-inline-math\">k<sub>d</sub>(b)</span> be the corresponding number of \\(b\\). Various generalizations of Alperin&#x27;s Weight Conjecture and McKay&#x27;s Conjecture are due to Reinhard Knorr, Geoffrey R. Robinson and Everett C. Dade. We follow Geoffrey R. Robinson&#x27;s approach to consider the Ordinary Weight Conjecture, and Dade&#x27;s Projective Conjecture. The general question is whether it follows from either of the latter two conjectures that <span class=\"etd-inline-math\">k<sub>d</sub>(B)=k<sub>d</sub>(b)</span> for all \\(d\\) for the \\(p\\)-block \\(B\\). The objective of this thesis is to show that these conjectures predict that <span class=\"etd-inline-math\">k<sub>d</sub>(B)=k<sub>d</sub>(b)</span>, for all non-negative integers \\(d\\). It is well known that <span class=\"etd-inline-math\">N<sub>G</sub>(E, b<sub>E</sub>)/EC<sub>G</sub>(E)</span> is a <span class=\"etd-inline-math\">p<sup>&#x27;</sup></span>-subgroup of the automorphism group of \\(E\\). Hence, we have considered some special cases of the above question.The unique largest normal \\(p\\)-subgroup of \\(G\\), <span class=\"etd-inline-math\">O<sub>p</sub>(G)</span> is the central focus of our attention. We consider the case that <span class=\"etd-inline-math\">O<sub>p</sub>(G)</span> is a central \\(p\\)-subgroup of \\(G\\), as well as the case that <span class=\"etd-inline-math\">O<sub>p</sub>(G)</span> is not central. In both cases, the common factor is that <span class=\"etd-inline-math\">O<sub>p</sub>(G)</span> is strictly contained in the defect group of \\(B\\).","abstract_has_math":true,"creators":["Alghamdi, Ahmad M."],"institution":"University of Birmingham","degree_name":"d_ph","degree_level":"d_ph","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2004,"date_issued":"2004","date_published":"2004","updated_at":"2026-07-24T01:10:41Z","subjects":["QA Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.sponsor","label":"Sponsor","values":["na"]},{"key":"dc:creator","label":"Author","values":["Alghamdi, Ahmad M."]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2004"]},{"key":"dc:date.issued","label":"Date","values":["2004"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["School of Mathematics & Statistics","School of Mathematics"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Birmingham"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["http://etheses.bham.ac.uk//id/eprint/86/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["d_ph"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["d_ph"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["QA Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://etheses.bham.ac.uk//id/eprint/86/1/Alghamdi04PhD.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let \\(p\\) be a rational odd prime number, \\(G\\) be a finite group such that \\(|G|=p^am\\), with \\(p \\nmid m\\). Let \\(B\\) be a \\(p\\)-block of \\(G\\) with a defect group \\(E\\) which is an extra-special \\(p\\)-group of order \\(p^3\\) and exponent \\(p\\). Consider a fixed maximal \\((G, B)\\)-subpair \\((E, b_E)\\). Let \\(b\\) be the Brauer correspondent of \\(B\\) for \\(N_G(E, b_E)\\). For a non-negative integer \\(d\\), let \\(k_d(B)\\) denote the number of irreducible characters \\(\\chi\\) in \\(B\\) which have \\(\\chi(1)_p=p^{a-d}\\) and let \\(k_d(b)\\) be the corresponding number of \\(b\\). Various generalizations of Alperin's Weight Conjecture and McKay's Conjecture are due to Reinhard Knorr, Geoffrey R. Robinson and Everett C. Dade. We follow Geoffrey R. Robinson's approach to consider the Ordinary Weight Conjecture, and Dade's Projective Conjecture. The general question is whether it follows from either of the latter two conjectures that \\(k_d(B)=k_d(b)\\) for all \\(d\\) for the \\(p\\)-block \\(B\\). The objective of this thesis is to show that these conjectures predict that \\(k_d(B)=k_d(b)\\), for all non-negative integers \\(d\\). It is well known that \\(N_G(E, b_E)/EC_G(E)\\) is a \\(p^'\\)-subgroup of the automorphism group of \\(E\\). Hence, we have considered some special cases of the above question.The unique largest normal \\(p\\)-subgroup of \\(G\\), \\(O_p(G)\\) is the central focus of our attention. We consider the case that \\(O_p(G)\\) is a central \\(p\\)-subgroup of \\(G\\), as well as the case that \\(O_p(G)\\) is not central. In both cases, the common factor is that \\(O_p(G)\\) is strictly contained in the defect group of \\(B\\)."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["The Ordinary Weight conjecture and Dade's Projective Conjecture for p-blocks with an extra-special defect group"]}]}],"canonical_facts":{"dc:contributor.sponsor":["na"],"dc:creator":["Alghamdi, Ahmad M."],"dc:date":["2004"],"dc:date.issued":["2004"],"dc:description.abstract":["Let \\(p\\) be a rational odd prime number, \\(G\\) be a finite group such that \\(|G|=p^am\\), with \\(p \\nmid m\\). Let \\(B\\) be a \\(p\\)-block of \\(G\\) with a defect group \\(E\\) which is an extra-special \\(p\\)-group of order \\(p^3\\) and exponent \\(p\\). Consider a fixed maximal \\((G, B)\\)-subpair \\((E, b_E)\\). Let \\(b\\) be the Brauer correspondent of \\(B\\) for \\(N_G(E, b_E)\\). For a non-negative integer \\(d\\), let \\(k_d(B)\\) denote the number of irreducible characters \\(\\chi\\) in \\(B\\) which have \\(\\chi(1)_p=p^{a-d}\\) and let \\(k_d(b)\\) be the corresponding number of \\(b\\). Various generalizations of Alperin's Weight Conjecture and McKay's Conjecture are due to Reinhard Knorr, Geoffrey R. Robinson and Everett C. Dade. We follow Geoffrey R. Robinson's approach to consider the Ordinary Weight Conjecture, and Dade's Projective Conjecture. The general question is whether it follows from either of the latter two conjectures that \\(k_d(B)=k_d(b)\\) for all \\(d\\) for the \\(p\\)-block \\(B\\). The objective of this thesis is to show that these conjectures predict that \\(k_d(B)=k_d(b)\\), for all non-negative integers \\(d\\). It is well known that \\(N_G(E, b_E)/EC_G(E)\\) is a \\(p^'\\)-subgroup of the automorphism group of \\(E\\). Hence, we have considered some special cases of the above question.The unique largest normal \\(p\\)-subgroup of \\(G\\), \\(O_p(G)\\) is the central focus of our attention. We consider the case that \\(O_p(G)\\) is a central \\(p\\)-subgroup of \\(G\\), as well as the case that \\(O_p(G)\\) is not central. In both cases, the common factor is that \\(O_p(G)\\) is strictly contained in the defect group of \\(B\\)."],"dc:format":["application/pdf"],"dc:identifier.uri":["http://etheses.bham.ac.uk//id/eprint/86/1/Alghamdi04PhD.pdf"],"dc:publisher.department":["School of Mathematics & Statistics","School of Mathematics"],"dc:publisher.institution":["University of Birmingham"],"dc:relation.isreferencedby":["http://etheses.bham.ac.uk//id/eprint/86/"],"dc:subject":["QA Mathematics"],"dc:title":["The Ordinary Weight conjecture and Dade's Projective Conjecture for p-blocks with an extra-special defect group"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["d_ph"],"dc:type.qualificationname":["d_ph"]},"updated_at":"2026-07-24T01:10:41Z"}