{"id":{"repo_id":"london-metro","oai_identifier":"oai:repository.londonmet.ac.uk:7533"},"canonical_url":"https://search.dev.ndltd.org/etd/london-metro/oai:repository.londonmet.ac.uk:7533","repository":{"repo_id":"london-metro","name":"London Metropolitan University","base_url":"https://repository.londonmet.ac.uk/cgi/oai2"},"display":{"title":"Geometric and homological methods in group theory: constructing small group resolutions","abstract":"Given two groups K and H for which we have the free crossed resolutions, B* --> K and C* --> H respectively. Our aim is to construct a free crossed resolution, A* --> G, by way of induction on the degree n, for any semidirect product G = K x H. First we show how to find a set Z1 of generators for the free group A1 and the corresponding unique epimorphism from the free group on those generators to the semidirect product. This gives us the I-dimensional free crossed resolution A1 --> G1 (see Proposition 4.1). Next we define a set of generators Z2 that together with Z1, constitute a generating set for the free crossed module A2 --> A1. where δ2 is crossed module homomorphism. Proposition 4.1 together with this free crossed module δ2 : A2 --> A1 define a 2-dimensional free crossed resolution for A2 --> A1 --> G (see Proposition 4.9). We then define an exact sequence A3 --> A2 --> A1, where A3, is an (A1/ δ2A2)-module on generating set Z3 with module homomorphism δ3 : A3 --> A2 defined on the generators. Proposition 4.11 says that we have a crossed complex of length 3, i.e., A3 --> A2 --> A1 --> G, where Imδ3 ⊆ Ker δ2.","abstract_html":"Given two groups K and H for which we have the free crossed resolutions, B* --> K and C* --> H respectively. Our aim is to construct a free crossed resolution, A* --> G, by way of induction on the degree n, for any semidirect product G = K x H. First we show how to find a set Z1 of generators for the free group A1 and the corresponding unique epimorphism from the free group on those generators to the semidirect product. This gives us the I-dimensional free crossed resolution A1 --> G1 (see Proposition 4.1). Next we define a set of generators Z2 that together with Z1, constitute a generating set for the free crossed module A2 --> A1. where δ2 is crossed module homomorphism. Proposition 4.1 together with this free crossed module δ2 : A2 --> A1 define a 2-dimensional free crossed resolution for A2 --> A1 --> G (see Proposition 4.9). We then define an exact sequence A3 --> A2 --> A1, where A3, is an (A1/ δ2A2)-module on generating set Z3 with module homomorphism δ3 : A3 --> A2 defined on the generators. Proposition 4.11 says that we have a crossed complex of length 3, i.e., A3 --> A2 --> A1 --> G, where Imδ3 ⊆ Ker δ2.","abstract_has_math":false,"creators":["Gill, Olivia Jo"],"institution":"London Metropolitan University","degree_name":"phd","degree_level":"doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011","date_published":"2011","updated_at":"2026-07-24T02:54:41Z","subjects":["510 Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier.grantnumber","label":"Dc Identifier Grantnumber","values":["N/A"],"render_values":[{"text":"N/A","href":null,"code":true}]}]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.sponsor","label":"Sponsor","values":["London Metropolitan University"]},{"key":"dc:creator","label":"Author","values":["Gill, Olivia Jo"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011"]},{"key":"dc:date.issued","label":"Date","values":["2011"]},{"key":"dc:publisher.department","label":"Dc Publisher Department","values":["School of Computing and Digital Media (SCDM)","Faculty of Life Sciences and Computing"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["London Metropolitan University"]},{"key":"dc:relation.isreferencedby","label":"Dc Relation Isreferencedby","values":["https://repository.londonmet.ac.uk/7533/"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["phd"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["510 Mathematics"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.grantnumber","label":"Dc Identifier Grantnumber","values":["N/A"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://repository.londonmet.ac.uk/7533/1/573402.pdf"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Given two groups K and H for which we have the free crossed resolutions, B* --> K and C* --> H respectively. Our aim is to construct a free crossed resolution, A* --> G, by way of induction on the degree n, for any semidirect product G = K x H. First we show how to find a set Z1 of generators for the free group A1 and the corresponding unique epimorphism from the free group on those generators to the semidirect product. This gives us the I-dimensional free crossed resolution A1 --> G1 (see Proposition 4.1). Next we define a set of generators Z2 that together with Z1, constitute a generating set for the free crossed module A2 --> A1. where δ2 is crossed module homomorphism. Proposition 4.1 together with this free crossed module δ2 : A2 --> A1 define a 2-dimensional free crossed resolution for A2 --> A1 --> G (see Proposition 4.9). We then define an exact sequence A3 --> A2 --> A1, where A3, is an (A1/ δ2A2)-module on generating set Z3 with module homomorphism δ3 : A3 --> A2 defined on the generators. Proposition 4.11 says that we have a crossed complex of length 3, i.e., A3 --> A2 --> A1 --> G, where Imδ3 ⊆ Ker δ2."]},{"key":"dc:format","label":"Dc Format","values":["text"]},{"key":"dc:title","label":"Title","values":["Geometric and homological methods in group theory: constructing small group resolutions"]}]}],"canonical_facts":{"dc:contributor.sponsor":["London Metropolitan University"],"dc:creator":["Gill, Olivia Jo"],"dc:date":["2011"],"dc:date.issued":["2011"],"dc:description.abstract":["Given two groups K and H for which we have the free crossed resolutions, B* --> K and C* --> H respectively. Our aim is to construct a free crossed resolution, A* --> G, by way of induction on the degree n, for any semidirect product G = K x H. First we show how to find a set Z1 of generators for the free group A1 and the corresponding unique epimorphism from the free group on those generators to the semidirect product. This gives us the I-dimensional free crossed resolution A1 --> G1 (see Proposition 4.1). Next we define a set of generators Z2 that together with Z1, constitute a generating set for the free crossed module A2 --> A1. where δ2 is crossed module homomorphism. Proposition 4.1 together with this free crossed module δ2 : A2 --> A1 define a 2-dimensional free crossed resolution for A2 --> A1 --> G (see Proposition 4.9). We then define an exact sequence A3 --> A2 --> A1, where A3, is an (A1/ δ2A2)-module on generating set Z3 with module homomorphism δ3 : A3 --> A2 defined on the generators. Proposition 4.11 says that we have a crossed complex of length 3, i.e., A3 --> A2 --> A1 --> G, where Imδ3 ⊆ Ker δ2."],"dc:format":["text"],"dc:identifier.grantnumber":["N/A"],"dc:identifier.uri":["https://repository.londonmet.ac.uk/7533/1/573402.pdf"],"dc:publisher.department":["School of Computing and Digital Media (SCDM)","Faculty of Life Sciences and Computing"],"dc:publisher.institution":["London Metropolitan University"],"dc:relation.isreferencedby":["https://repository.londonmet.ac.uk/7533/"],"dc:subject":["510 Mathematics"],"dc:title":["Geometric and homological methods in group theory: constructing small group resolutions"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["doctoral"],"dc:type.qualificationname":["phd"]},"updated_at":"2026-07-24T02:54:41Z"}