{"id":{"repo_id":"regina","oai_identifier":"oai:uregina.scholaris.ca:10294/16478"},"canonical_url":"https://search.dev.ndltd.org/etd/regina/oai:uregina.scholaris.ca:10294/16478","repository":{"repo_id":"regina","name":"University of Regina","base_url":"https://uregina.scholaris.ca/server/oai/request"},"display":{"title":"The Macdonald group","abstract":"Given α ∈ Z, the Macdonald group G(α) is defined by G(α) = ⟨ A,B | A[A,B] = Aα, B[B,A] = Bα ⟩. It is known that G(α) is finite if and only if α ̸= 1, in which case the prime factors of |G(α)| are those of α − 1. It is also known that G(α) is nilpotent in certain cases. We show that G(α) is always nilpotent, so that for α ̸= 1, G(α) is the direct product of its Sylow subgroups. In the first third of the thesis, we determine the order, upper and lower central series, nilpotency class, and exponent of each of these Sylow subgroups. For the remaining two thirds of the thesis we concentrate on the Sylow 2-subgroup J = J(α) of G(α), so we assume that α = 1 + 2mℓ, where m ≥ 1 and ℓ is odd. We show that J has presentation J = ⟨ x, y | x[x,y] = x1+2mℓ, y[y,x] = y1+2mℓ, x23m−1 = 1 = y23m−1⟩, order 27m−3, and nilpotency class 5 if m > 1 and 3 if m = 1. In the middle third of the thesis, we determine the automorphism groups of the 2-groups J, H = J/Z(J) and K = H/Z(H), where |H| = 26m−3 and |K| = 25m−3. Explicit multiplication, power, and commutator formulas for J, H, and K are given, and used in the calculation of Aut(J), Aut(H), and Aut(K). In the final third of the thesis, we consider the infinite family of finite 2-groups {J(α)}α̸=1 and settle the following isomorphism problem: given α ̸= 1 ̸= α′ ∈ Z, when are J(α) and J(α′) isomorphic?","abstract_html":"Given α ∈ Z, the Macdonald group G(α) is defined by G(α) = ⟨ A,B | A[A,B] = Aα, B[B,A] = Bα ⟩. It is known that G(α) is finite if and only if α ̸= 1, in which case the prime factors of |G(α)| are those of α − 1. It is also known that G(α) is nilpotent in certain cases. We show that G(α) is always nilpotent, so that for α ̸= 1, G(α) is the direct product of its Sylow subgroups. In the first third of the thesis, we determine the order, upper and lower central series, nilpotency class, and exponent of each of these Sylow subgroups. For the remaining two thirds of the thesis we concentrate on the Sylow 2-subgroup J = J(α) of G(α), so we assume that α = 1 + 2mℓ, where m ≥ 1 and ℓ is odd. We show that J has presentation J = ⟨ x, y | x[x,y] = x1+2mℓ, y[y,x] = y1+2mℓ, x23m−1 = 1 = y23m−1⟩, order 27m−3, and nilpotency class 5 if m > 1 and 3 if m = 1. In the middle third of the thesis, we determine the automorphism groups of the 2-groups J, H = J/Z(J) and K = H/Z(H), where |H| = 26m−3 and |K| = 25m−3. Explicit multiplication, power, and commutator formulas for J, H, and K are given, and used in the calculation of Aut(J), Aut(H), and Aut(K). In the final third of the thesis, we consider the infinite family of finite 2-groups {J(α)}α̸=1 and settle the following isomorphism problem: given α ̸= 1 ̸= α′ ∈ Z, when are J(α) and J(α′) isomorphic?","abstract_has_math":false,"creators":["Montoya Ocampo, Alexander"],"institution":"Faculty of Graduate Studies and Research, University of Regina","degree_name":"Master of Science (MSc)","degree_level":"Master's","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Szechtman, Fernando"],"committee_chairs":[],"committee_members":["Herman, Allen","Gilligan, Bruce"],"year":2024,"date_issued":"2024-02","date_published":"2024-02","updated_at":"2026-07-24T04:03:30Z","subjects":[],"languages":["en"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.82465/4047"],"render_values":[{"text":"https://doi.org/10.82465/4047","href":"https://doi.org/10.82465/4047","code":true}]}]},"links":{"outbound_url":"https://hdl.handle.net/10294/16478","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Szechtman, Fernando"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Herman, Allen","Gilligan, Bruce"]},{"key":"dc:creator","label":"Author","values":["Montoya Ocampo, Alexander"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2024-10-11T20:12:39Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2024-10-11T20:12:39Z"]},{"key":"dc:date.issued","label":"Date","values":["2024-02"]},{"key":"dc:publisher","label":"Institution","values":["Faculty of Graduate Studies and Research, University of Regina"]},{"key":"dc:type","label":"Dc Type","values":["master thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Master's"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MSc)"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["Faculty of Graduate Studies and Research, University of Regina"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["en"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["https://doi.org/10.82465/4047"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10294/16478"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics, University of Regina. v, 113 p."]},{"key":"dc:description.abstract","label":"Abstract","values":["Given α ∈ Z, the Macdonald group G(α) is defined by G(α) = ⟨ A,B | A[A,B] = Aα, B[B,A] = Bα ⟩. It is known that G(α) is finite if and only if α ̸= 1, in which case the prime factors of |G(α)| are those of α − 1. It is also known that G(α) is nilpotent in certain cases. We show that G(α) is always nilpotent, so that for α ̸= 1, G(α) is the direct product of its Sylow subgroups. In the first third of the thesis, we determine the order, upper and lower central series, nilpotency class, and exponent of each of these Sylow subgroups. For the remaining two thirds of the thesis we concentrate on the Sylow 2-subgroup J = J(α) of G(α), so we assume that α = 1 + 2mℓ, where m ≥ 1 and ℓ is odd. We show that J has presentation J = ⟨ x, y | x[x,y] = x1+2mℓ, y[y,x] = y1+2mℓ, x23m−1 = 1 = y23m−1⟩, order 27m−3, and nilpotency class 5 if m > 1 and 3 if m = 1. In the middle third of the thesis, we determine the automorphism groups of the 2-groups J, H = J/Z(J) and K = H/Z(H), where |H| = 26m−3 and |K| = 25m−3. Explicit multiplication, power, and commutator formulas for J, H, and K are given, and used in the calculation of Aut(J), Aut(H), and Aut(K). In the final third of the thesis, we consider the infinite family of finite 2-groups {J(α)}α̸=1 and settle the following isomorphism problem: given α ̸= 1 ̸= α′ ∈ Z, when are J(α) and J(α′) isomorphic?"]},{"key":"dc:title","label":"Title","values":["The Macdonald group"]}]}],"canonical_facts":{"dc:contributor.advisor":["Szechtman, Fernando"],"dc:contributor.committeemember":["Herman, Allen","Gilligan, Bruce"],"dc:creator":["Montoya Ocampo, Alexander"],"dc:date.accessioned":["2024-10-11T20:12:39Z"],"dc:date.available":["2024-10-11T20:12:39Z"],"dc:date.issued":["2024-02"],"dc:description":["A Thesis Submitted to the Faculty of Graduate Studies and Research In Partial Fulfillment of the Requirements for the Degree of Master of Science in Mathematics, University of Regina. v, 113 p."],"dc:description.abstract":["Given α ∈ Z, the Macdonald group G(α) is defined by G(α) = ⟨ A,B | A[A,B] = Aα, B[B,A] = Bα ⟩. It is known that G(α) is finite if and only if α ̸= 1, in which case the prime factors of |G(α)| are those of α − 1. It is also known that G(α) is nilpotent in certain cases. We show that G(α) is always nilpotent, so that for α ̸= 1, G(α) is the direct product of its Sylow subgroups. In the first third of the thesis, we determine the order, upper and lower central series, nilpotency class, and exponent of each of these Sylow subgroups. For the remaining two thirds of the thesis we concentrate on the Sylow 2-subgroup J = J(α) of G(α), so we assume that α = 1 + 2mℓ, where m ≥ 1 and ℓ is odd. We show that J has presentation J = ⟨ x, y | x[x,y] = x1+2mℓ, y[y,x] = y1+2mℓ, x23m−1 = 1 = y23m−1⟩, order 27m−3, and nilpotency class 5 if m > 1 and 3 if m = 1. In the middle third of the thesis, we determine the automorphism groups of the 2-groups J, H = J/Z(J) and K = H/Z(H), where |H| = 26m−3 and |K| = 25m−3. Explicit multiplication, power, and commutator formulas for J, H, and K are given, and used in the calculation of Aut(J), Aut(H), and Aut(K). In the final third of the thesis, we consider the infinite family of finite 2-groups {J(α)}α̸=1 and settle the following isomorphism problem: given α ̸= 1 ̸= α′ ∈ Z, when are J(α) and J(α′) isomorphic?"],"dc:identifier.doi":["https://doi.org/10.82465/4047"],"dc:identifier.uri":["https://hdl.handle.net/10294/16478"],"dc:language.iso":["en"],"dc:publisher":["Faculty of Graduate Studies and Research, University of Regina"],"dc:title":["The Macdonald group"],"dc:type":["master thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Master's"],"thesis:degree_name":["Master of Science (MSc)"],"thesis:institution_name":["Faculty of Graduate Studies and Research, University of Regina"]},"updated_at":"2026-07-24T04:03:30Z"}