{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/68170"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/68170","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Lifting Automorphisms to Stem Extensions of a Finite Group","abstract":"A central extension of finite groups e: 0 (--->) A (--->) E (--->) G (--->)1 is said to be a stem extension of G if A is contained in the commutator subgroup E' of E. Schur showed that A must be isomorphic to a subgroup of the finite abelian group M = M(G) = H('2)(G,(//C)('(.))), where ((//C)('(.))) is the group of complex units. In case A (TURNEQ) M, we say that e is a stem cover of G.","abstract_html":"A central extension of finite groups e: 0 (--->) A (--->) E (--->) G (--->)1 is said to be a stem extension of G if A is contained in the commutator subgroup E' of E. Schur showed that A must be isomorphic to a subgroup of the finite abelian group M = M(G) = H('2)(G,(//C)('(.))), where ((//C)('(.))) is the group of complex units. In case A (TURNEQ) M, we say that e is a stem cover of G.","abstract_has_math":false,"creators":["Fry, Michael Donoho"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2014,"date_issued":"2014-12-14T13:09:46Z","date_published":"2014-12-14T13:09:46Z","updated_at":"2026-07-22T22:25:58Z","subjects":["Mathematics"],"languages":["eng"],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8026495"],"render_values":[{"text":"(UMI)AAI8026495","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/68170","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Fry, Michael Donoho"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2014-12-14T13:09:46Z","10000-01-01","1980"]},{"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"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/68170","(UMI)AAI8026495"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["A central extension of finite groups e: 0 (--->) A (--->) E (--->) G (--->)1 is said to be a stem extension of G if A is contained in the commutator subgroup E' of E. Schur showed that A must be isomorphic to a subgroup of the finite abelian group M = M(G) = H('2)(G,(//C)('(.))), where ((//C)('(.))) is the group of complex units. In case A (TURNEQ) M, we say that e is a stem cover of G.","An automorphism (sigma) of G lifts to e if there is a commutativediagram e: 0 (--->) A (--->) E (--->) G (--->) 1 (TURNEQ)(DARR) (TURNEQ)(DARR) (DARR)(sigma) e: 0 (--->) A (--->) E (--->) G (--->) 1. The group G is called an group if there is some stem cover of G to which every automorphism of G lifts. One known fact is that if GCD( (VBAR)G/G'(VBAR), (VBAR)M(VBAR)) = 1, then G is . Theorem. (I) If every Sylow subgroup of G is , then G is . (II) If for each prime p dividing GCD( (VBAR)G/G'(VBAR), (VBAR)M(VBAR) , (VBAR)Out G(VBAR) ), some Sylow p-subgroup of Aut G lifts to some stem cover of G, then G is . (Out G is the outer automorphism group of G.) (III) If G is abelian of odd order, then G is . (IV) Suppose G is elementary of order 2('r). Then G is if and only if r (LESSTHEQ) 2.","What sets groups apartfrom non-groups? Two extensions e(,1) and e(,2) of G are isomorphic if there is a diagram e(,1): 0 (--->) A(,1) (--->) E(,1) (--->) G (--->) 1 (TURNEQ)(DARR) (TURNEQ)(DARR) (TURNEQ)(DARR) e(,2): 0 (--->) A(,2) (--->) E(,2) (--->) G (--->) 1.","In case the right-hand map is 1, we say that e(,1) and e(,2)are type 1 isomorphic. The set (GAMMA) of type 1 isomorphism classes of stem covers of G can be made into an Aut G-set in such a way that two stem covers are isomorphic (as extensions) if and only if the corresponding elements of (GAMMA) are in the same Aut G-orbit. Theorem. The finite group G is if and only if the Aut G-set (GAMMA) has a fixed point. If G is , then (GAMMA) can be given an additive structure making it an Aut G-module. In fact, there is a canonical way of making the group K = Ext(G/G',M('*)) into an Aut G-module (M('*) = Hom(M,(//C)('(.)))) and if G is , then (GAMMA) and K are equivalent Aut G-sets.","The property is also characterized by the existence of a "nice"splitting of the Universal Coefficient Sequence 0 (--->) Ext (G/G',M('*)) (--->) H('2)(G,M('*)) (--->) Hom(M('**),M) (--->) 0. Somewhat related to the lifting problem is the notion of isoclinism of extensions. For a given central expansion e: 0 (--->) A (--->) E (--->) G (--->) 1, let U(e) denote the subgroup of Aut G consisting of the autoclinisms of e (isoclinisms of e to itself). If I(,e) denotes the group of automorphisms of G that lift to e, then we obtain a tower I(,e) (LESSTHEQ) U(e) (LESSTHEQ) Aut G.","It is known that everycentral extension of G is isoclinic to some stem extension of G. Theorem. The finite group G is if and only if every central extension e of G is isoclinic to some stem extension e' of G satisfying I(,e)' = U(e') = U(e).","Made available in DSpace on 2014-12-14T13:09:46Z (GMT). No. of bitstreams: 1 8026495.pdf: 2415849 bytes, checksum: 2970f14c189760ad99837cd2e39c935a (MD5) Previous issue date: 1980","Embargo set by: Seth Robbins for item 68348 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs","U of I Only","112 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980."]},{"key":"dc:title","label":"Title","values":["Lifting Automorphisms to Stem Extensions of a Finite Group"]}]}],"canonical_facts":{"dc:creator":["Fry, Michael Donoho"],"dc:date":["2014-12-14T13:09:46Z","10000-01-01","1980"],"dc:description":["A central extension of finite groups e: 0 (--->) A (--->) E (--->) G (--->)1 is said to be a stem extension of G if A is contained in the commutator subgroup E' of E. Schur showed that A must be isomorphic to a subgroup of the finite abelian group M = M(G) = H('2)(G,(//C)('(.))), where ((//C)('(.))) is the group of complex units. In case A (TURNEQ) M, we say that e is a stem cover of G.","An automorphism (sigma) of G lifts to e if there is a commutativediagram e: 0 (--->) A (--->) E (--->) G (--->) 1 (TURNEQ)(DARR) (TURNEQ)(DARR) (DARR)(sigma) e: 0 (--->) A (--->) E (--->) G (--->) 1. The group G is called an group if there is some stem cover of G to which every automorphism of G lifts. One known fact is that if GCD( (VBAR)G/G'(VBAR), (VBAR)M(VBAR)) = 1, then G is . Theorem. (I) If every Sylow subgroup of G is , then G is . (II) If for each prime p dividing GCD( (VBAR)G/G'(VBAR), (VBAR)M(VBAR) , (VBAR)Out G(VBAR) ), some Sylow p-subgroup of Aut G lifts to some stem cover of G, then G is . (Out G is the outer automorphism group of G.) (III) If G is abelian of odd order, then G is . (IV) Suppose G is elementary of order 2('r). Then G is if and only if r (LESSTHEQ) 2.","What sets groups apartfrom non-groups? Two extensions e(,1) and e(,2) of G are isomorphic if there is a diagram e(,1): 0 (--->) A(,1) (--->) E(,1) (--->) G (--->) 1 (TURNEQ)(DARR) (TURNEQ)(DARR) (TURNEQ)(DARR) e(,2): 0 (--->) A(,2) (--->) E(,2) (--->) G (--->) 1.","In case the right-hand map is 1, we say that e(,1) and e(,2)are type 1 isomorphic. The set (GAMMA) of type 1 isomorphism classes of stem covers of G can be made into an Aut G-set in such a way that two stem covers are isomorphic (as extensions) if and only if the corresponding elements of (GAMMA) are in the same Aut G-orbit. Theorem. The finite group G is if and only if the Aut G-set (GAMMA) has a fixed point. If G is , then (GAMMA) can be given an additive structure making it an Aut G-module. In fact, there is a canonical way of making the group K = Ext(G/G',M('*)) into an Aut G-module (M('*) = Hom(M,(//C)('(.)))) and if G is , then (GAMMA) and K are equivalent Aut G-sets.","The property is also characterized by the existence of a "nice"splitting of the Universal Coefficient Sequence 0 (--->) Ext (G/G',M('*)) (--->) H('2)(G,M('*)) (--->) Hom(M('**),M) (--->) 0. Somewhat related to the lifting problem is the notion of isoclinism of extensions. For a given central expansion e: 0 (--->) A (--->) E (--->) G (--->) 1, let U(e) denote the subgroup of Aut G consisting of the autoclinisms of e (isoclinisms of e to itself). If I(,e) denotes the group of automorphisms of G that lift to e, then we obtain a tower I(,e) (LESSTHEQ) U(e) (LESSTHEQ) Aut G.","It is known that everycentral extension of G is isoclinic to some stem extension of G. Theorem. The finite group G is if and only if every central extension e of G is isoclinic to some stem extension e' of G satisfying I(,e)' = U(e') = U(e).","Made available in DSpace on 2014-12-14T13:09:46Z (GMT). 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