{"id":{"repo_id":"birmingham","oai_identifier":"oai:etheses.bham.ac.uk:559"},"canonical_url":"https://search.dev.ndltd.org/etd/birmingham/oai:etheses.bham.ac.uk:559","repository":{"repo_id":"birmingham","name":"University of Birmingham","base_url":"https://etheses.bham.ac.uk/cgi/oai2"},"display":{"title":"Universal completions of the cyclic amalgams of the same type","abstract":"Automorphisms of Z/nZ is isomorphic to (Z/Z)×. If G is a finite abelian group, which is isomorphic to direct product of m cyclic groups of order q where q = p\\(^n\\) for some prime p. Then Aut(G) is isomorphic to the set of m×m matrices with determinant coprime to p, GL\\(_m\\) (e\\(_q\\)) Also Aut(G)=p\\(^{(n-1)}\\) \\(^{(m2)}\\) GL\\(_m\\) (Z\\(_q\\)). If \\(\\alpha\\)is an automorphism of S\\(_n\\)and t is a transposition of S\\(_n\\)for n \\(\\not\\) 6, then \\(\\alpha\\)(t)is a transposition. If \\(\\alpha\\) maps transposition to a transposition, then \\(\\alpha\\) is an inner automorphism. Then AUT (S\\(_n)\\) \\(\\simeq\\) S \\(_n\\) \\(\\not\\) 6. Furthermore, there exists an outer automorphism of S\\(_6\\)and OUT (S\\(_6\\)) \\(\\simeq\\) \\(\\frac{z}{zz}\\). Thus OUT (S\\(_g\\))=2. Coset enumeration is one of the basic methods for investigating finitely generated subgroups in finitely presented.. Information are gradually added to a coset, a relation, a subgroup tables and once they are filled in, all cosets have been enumerated, the algorithm terminates. Goldschmidt’s Lemma on the number of isomorphism classes of amalgams having fixed type, verify that there is one isomorphism class of amalgam of type A=(S\\(_n\\)S\\(_n\\)S_(n-1), \\(\\phi\\)\\(_1\\), \\(\\phi\\)\\(_2\\)) where \\(\\phi\\) is an identity map from S_(n-1) to (S\\(_n\\)for i=1, 2 and n\\(\\not\\) 2,3,6,7. When n=2,7 we have two isomorphic class of amalgam of type A. Finally, i A and A’ are cyclic amalgams of the same type then there universal completions are isospectral.","abstract_html":"Automorphisms of Z/nZ is isomorphic to (Z/Z)×. If G is a finite abelian group, which is isomorphic to direct product of m cyclic groups of order q where q = p<span class=\"etd-inline-math\"><sup>n</sup></span> for some prime p. Then Aut(G) is isomorphic to the set of m×m matrices with determinant coprime to p, GL<span class=\"etd-inline-math\"><sub>m</sub></span> (e<span class=\"etd-inline-math\"><sub>q</sub></span>) Also Aut(G)=p<span class=\"etd-inline-math\"><sup>(n-1)</sup></span> <span class=\"etd-inline-math\"><sup>(m2)</sup></span> GL<span class=\"etd-inline-math\"><sub>m</sub></span> (Z<span class=\"etd-inline-math\"><sub>q</sub></span>). If <span class=\"etd-inline-math\">&alpha;</span>is an automorphism of S<span class=\"etd-inline-math\"><sub>n</sub></span>and t is a transposition of S<span class=\"etd-inline-math\"><sub>n</sub></span>for n \\(\\not\\) 6, then <span class=\"etd-inline-math\">&alpha;</span>(t)is a transposition. If <span class=\"etd-inline-math\">&alpha;</span> maps transposition to a transposition, then <span class=\"etd-inline-math\">&alpha;</span> is an inner automorphism. Then AUT (S<span class=\"etd-inline-math\"><sub>n</sub>)</span> \\(\\simeq\\) S <span class=\"etd-inline-math\"><sub>n</sub></span> \\(\\not\\) 6. Furthermore, there exists an outer automorphism of S<span class=\"etd-inline-math\"><sub>6</sub></span>and OUT (S<span class=\"etd-inline-math\"><sub>6</sub></span>) \\(\\simeq\\) \\(\\frac{z}{zz}\\). Thus OUT (S<span class=\"etd-inline-math\"><sub>g</sub></span>)=2. Coset enumeration is one of the basic methods for investigating finitely generated subgroups in finitely presented.. Information are gradually added to a coset, a relation, a subgroup tables and once they are filled in, all cosets have been enumerated, the algorithm terminates. Goldschmidt’s Lemma on the number of isomorphism classes of amalgams having fixed type, verify that there is one isomorphism class of amalgam of type A=(S<span class=\"etd-inline-math\"><sub>n</sub></span>S<span class=\"etd-inline-math\"><sub>n</sub></span>S_(n-1), \\(\\phi\\)<span class=\"etd-inline-math\"><sub>1</sub></span>, \\(\\phi\\)<span class=\"etd-inline-math\"><sub>2</sub></span>) where \\(\\phi\\) is an identity map from S_(n-1) to (S<span class=\"etd-inline-math\"><sub>n</sub></span>for i=1, 2 and n\\(\\not\\) 2,3,6,7. When n=2,7 we have two isomorphic class of amalgam of type A. Finally, i A and A’ are cyclic amalgams of the same type then there universal completions are isospectral.","abstract_has_math":true,"creators":["Atapattu Arachchille, Kanchana Chamila"],"institution":"University of Birmingham","degree_name":"m_ph","degree_level":"m_ph","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2010,"date_issued":"2010-07","date_published":"2010-07","updated_at":"2026-07-24T01:11:02Z","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":["other"]},{"key":"dc:creator","label":"Author","values":["Atapattu Arachchille, Kanchana Chamila"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2010-07"]},{"key":"dc:date.issued","label":"Date","values":["2010-07"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["College of Engineering & Physical Sciences","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/559/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["m_ph"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["m_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/559/4/arachchille10MPhil.pdf","http://etheses.bham.ac.uk//id/eprint/559/3/Decl_IS_Arachchille.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Automorphisms of Z/nZ is isomorphic to (Z/Z)×. If G is a finite abelian group, which is isomorphic to direct product of m cyclic groups of order q where q = p\\(^n\\) for some prime p. Then Aut(G) is isomorphic to the set of m×m matrices with determinant coprime to p, GL\\(_m\\) (e\\(_q\\)) Also Aut(G)=p\\(^{(n-1)}\\) \\(^{(m2)}\\) GL\\(_m\\) (Z\\(_q\\)). If \\(\\alpha\\)is an automorphism of S\\(_n\\)and t is a transposition of S\\(_n\\)for n \\(\\not\\) 6, then \\(\\alpha\\)(t)is a transposition. If \\(\\alpha\\) maps transposition to a transposition, then \\(\\alpha\\) is an inner automorphism. Then AUT (S\\(_n)\\) \\(\\simeq\\) S \\(_n\\) \\(\\not\\) 6. Furthermore, there exists an outer automorphism of S\\(_6\\)and OUT (S\\(_6\\)) \\(\\simeq\\) \\(\\frac{z}{zz}\\). Thus OUT (S\\(_g\\))=2. Coset enumeration is one of the basic methods for investigating finitely generated subgroups in finitely presented.. Information are gradually added to a coset, a relation, a subgroup tables and once they are filled in, all cosets have been enumerated, the algorithm terminates. Goldschmidt’s Lemma on the number of isomorphism classes of amalgams having fixed type, verify that there is one isomorphism class of amalgam of type A=(S\\(_n\\)S\\(_n\\)S_(n-1), \\(\\phi\\)\\(_1\\), \\(\\phi\\)\\(_2\\)) where \\(\\phi\\) is an identity map from S_(n-1) to (S\\(_n\\)for i=1, 2 and n\\(\\not\\) 2,3,6,7. When n=2,7 we have two isomorphic class of amalgam of type A. Finally, i A and A’ are cyclic amalgams of the same type then there universal completions are isospectral."]},{"key":"dc:format","label":"Dc Format","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Universal completions of the cyclic amalgams of the same type"]}]}],"canonical_facts":{"dc:contributor.sponsor":["other"],"dc:creator":["Atapattu Arachchille, Kanchana Chamila"],"dc:date":["2010-07"],"dc:date.issued":["2010-07"],"dc:description.abstract":["Automorphisms of Z/nZ is isomorphic to (Z/Z)×. If G is a finite abelian group, which is isomorphic to direct product of m cyclic groups of order q where q = p\\(^n\\) for some prime p. Then Aut(G) is isomorphic to the set of m×m matrices with determinant coprime to p, GL\\(_m\\) (e\\(_q\\)) Also Aut(G)=p\\(^{(n-1)}\\) \\(^{(m2)}\\) GL\\(_m\\) (Z\\(_q\\)). If \\(\\alpha\\)is an automorphism of S\\(_n\\)and t is a transposition of S\\(_n\\)for n \\(\\not\\) 6, then \\(\\alpha\\)(t)is a transposition. If \\(\\alpha\\) maps transposition to a transposition, then \\(\\alpha\\) is an inner automorphism. Then AUT (S\\(_n)\\) \\(\\simeq\\) S \\(_n\\) \\(\\not\\) 6. Furthermore, there exists an outer automorphism of S\\(_6\\)and OUT (S\\(_6\\)) \\(\\simeq\\) \\(\\frac{z}{zz}\\). Thus OUT (S\\(_g\\))=2. Coset enumeration is one of the basic methods for investigating finitely generated subgroups in finitely presented.. Information are gradually added to a coset, a relation, a subgroup tables and once they are filled in, all cosets have been enumerated, the algorithm terminates. Goldschmidt’s Lemma on the number of isomorphism classes of amalgams having fixed type, verify that there is one isomorphism class of amalgam of type A=(S\\(_n\\)S\\(_n\\)S_(n-1), \\(\\phi\\)\\(_1\\), \\(\\phi\\)\\(_2\\)) where \\(\\phi\\) is an identity map from S_(n-1) to (S\\(_n\\)for i=1, 2 and n\\(\\not\\) 2,3,6,7. When n=2,7 we have two isomorphic class of amalgam of type A. Finally, i A and A’ are cyclic amalgams of the same type then there universal completions are isospectral."],"dc:format":["application/pdf"],"dc:identifier.uri":["http://etheses.bham.ac.uk//id/eprint/559/4/arachchille10MPhil.pdf","http://etheses.bham.ac.uk//id/eprint/559/3/Decl_IS_Arachchille.pdf"],"dc:publisher.department":["College of Engineering & Physical Sciences","School of Mathematics"],"dc:publisher.institution":["University of Birmingham"],"dc:relation.isreferencedby":["http://etheses.bham.ac.uk//id/eprint/559/"],"dc:subject":["QA Mathematics"],"dc:title":["Universal completions of the cyclic amalgams of the same type"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["m_ph"],"dc:type.qualificationname":["m_ph"]},"updated_at":"2026-07-24T01:11:02Z"}