{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/20160"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/20160","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Finite groups with a special 2-generator property, and order of centralizers in finite groups","abstract":"\"This paper deals with finite groups, and has two parts. In part I J. L. Brenner and James Wielgold (I,3) defined a finite nonabelian group G as lying in $\\Gamma\\sb1\\sp{(2)}$ (spread one-two) if for every 1 $\\not=$ x $\\in$ G, either x is an involution and G = $\\langle$x,y$\\rangle$ for some y $\\in$ G or x is not an involution and there is an involution z $\\in$ G with G = $\\langle$x,z$\\rangle$. We show that \"\"most\"\" of the simple groups of Lie type do not lie in $\\Gamma\\sb1\\sp{(2)}$, we classify all those solvable groups which lie in $\\Gamma\\sb1\\sp{(2)}$, and we show that a finite non-simple non-solvable group lies in $\\Gamma\\sb1\\sp{(2)}$ if it is isomorphic to the semi-direct product of N and $\\langle$x$\\rangle$ where x is an involution and N is a simple nonabelian group. Many simple groups are excluded from being candidates for the N above.\"","abstract_html":"&quot;This paper deals with finite groups, and has two parts. In part I J. L. Brenner and James Wielgold (I,3) defined a finite nonabelian group G as lying in $\\Gamma\\sb1\\sp{(2)}$ (spread one-two) if for every 1 $\\not=$ x $\\in$ G, either x is an involution and G = $\\langle$x,y$\\rangle$ for some y $\\in$ G or x is not an involution and there is an involution z $\\in$ G with G = $\\langle$x,z$\\rangle$. We show that &quot;&quot;most&quot;&quot; of the simple groups of Lie type do not lie in $\\Gamma\\sb1\\sp{(2)}$, we classify all those solvable groups which lie in $\\Gamma\\sb1\\sp{(2)}$, and we show that a finite non-simple non-solvable group lies in $\\Gamma\\sb1\\sp{(2)}$ if it is isomorphic to the semi-direct product of N and $\\langle$x$\\rangle$ where x is an involution and N is a simple nonabelian group. Many simple groups are excluded from being candidates for the N above.&quot;","abstract_has_math":true,"creators":["Foguel, Tuval Shmuel"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Rotman, Joseph J."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T12:30:43Z","date_published":"2011-05-07T12:30:43Z","updated_at":"2026-07-22T22:25:15Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1992 Foguel, Tuval Shmuel"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9236461","(UMI)AAI9236461"],"render_values":[{"text":"AAI9236461","href":null,"code":true},{"text":"(UMI)AAI9236461","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/20160","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Rotman, Joseph J."]},{"key":"dc:creator","label":"Author","values":["Foguel, Tuval Shmuel"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T12:30:43Z","10000-01-01","1992"]},{"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"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1992 Foguel, Tuval Shmuel"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9236461","(UMI)AAI9236461","http://hdl.handle.net/2142/20160"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["\"This paper deals with finite groups, and has two parts. In part I J. L. Brenner and James Wielgold (I,3) defined a finite nonabelian group G as lying in $\\Gamma\\sb1\\sp{(2)}$ (spread one-two) if for every 1 $\\not=$ x $\\in$ G, either x is an involution and G = $\\langle$x,y$\\rangle$ for some y $\\in$ G or x is not an involution and there is an involution z $\\in$ G with G = $\\langle$x,z$\\rangle$. We show that \"\"most\"\" of the simple groups of Lie type do not lie in $\\Gamma\\sb1\\sp{(2)}$, we classify all those solvable groups which lie in $\\Gamma\\sb1\\sp{(2)}$, and we show that a finite non-simple non-solvable group lies in $\\Gamma\\sb1\\sp{(2)}$ if it is isomorphic to the semi-direct product of N and $\\langle$x$\\rangle$ where x is an involution and N is a simple nonabelian group. Many simple groups are excluded from being candidates for the N above.\"","Part II includes a characterization of all groups G having a subgroup A with $\\vert$A$\\Vert$C$\\sb{\\rm G}$(A)$\\vert$ $>$ $\\vert$G$\\vert$, and those for which m$\\sb1$ = sup $\\{\\vert$B$\\Vert$C$\\sb{\\rm G}$(B)$\\vert$: B $\\le$ G$\\}$ = $\\vert$G$\\vert$. It is shown also that if G is not a direct product, then either there exists a nontrivial characteristic abelian subgroup A of G with $\\vert$A$\\Vert$C$\\sb{\\rm G}$(A)$\\vert$ $\\ge$ $\\vert$G$\\vert$, or $\\vert$B$\\Vert$C$\\sb{\\rm G}$(B)$\\vert$ $<$ $\\vert$G$\\vert$ for any proper nontrivial subgroup B of G.","Made available in DSpace on 2011-05-07T12:30:43Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9236461.pdf: 1841647 bytes, checksum: 0b79a8c8d1588eea079c69288eb5e206 (MD5) Previous issue date: 1992","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:41:58Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:18:13-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"]},{"key":"dc:title","label":"Title","values":["Finite groups with a special 2-generator property, and order of centralizers in finite groups"]}]}],"canonical_facts":{"dc:contributor":["Rotman, Joseph J."],"dc:creator":["Foguel, Tuval Shmuel"],"dc:date":["2011-05-07T12:30:43Z","10000-01-01","1992"],"dc:description":["\"This paper deals with finite groups, and has two parts. In part I J. L. Brenner and James Wielgold (I,3) defined a finite nonabelian group G as lying in $\\Gamma\\sb1\\sp{(2)}$ (spread one-two) if for every 1 $\\not=$ x $\\in$ G, either x is an involution and G = $\\langle$x,y$\\rangle$ for some y $\\in$ G or x is not an involution and there is an involution z $\\in$ G with G = $\\langle$x,z$\\rangle$. We show that \"\"most\"\" of the simple groups of Lie type do not lie in $\\Gamma\\sb1\\sp{(2)}$, we classify all those solvable groups which lie in $\\Gamma\\sb1\\sp{(2)}$, and we show that a finite non-simple non-solvable group lies in $\\Gamma\\sb1\\sp{(2)}$ if it is isomorphic to the semi-direct product of N and $\\langle$x$\\rangle$ where x is an involution and N is a simple nonabelian group. Many simple groups are excluded from being candidates for the N above.\"","Part II includes a characterization of all groups G having a subgroup A with $\\vert$A$\\Vert$C$\\sb{\\rm G}$(A)$\\vert$ $>$ $\\vert$G$\\vert$, and those for which m$\\sb1$ = sup $\\{\\vert$B$\\Vert$C$\\sb{\\rm G}$(B)$\\vert$: B $\\le$ G$\\}$ = $\\vert$G$\\vert$. It is shown also that if G is not a direct product, then either there exists a nontrivial characteristic abelian subgroup A of G with $\\vert$A$\\Vert$C$\\sb{\\rm G}$(A)$\\vert$ $\\ge$ $\\vert$G$\\vert$, or $\\vert$B$\\Vert$C$\\sb{\\rm G}$(B)$\\vert$ $<$ $\\vert$G$\\vert$ for any proper nontrivial subgroup B of G.","Made available in DSpace on 2011-05-07T12:30:43Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9236461.pdf: 1841647 bytes, checksum: 0b79a8c8d1588eea079c69288eb5e206 (MD5) Previous issue date: 1992","Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:41:58Z Item is restricted indefinitely.","Restriction data tranferred 2014-07-01T11:18:13-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission","ETDs are only available to UIUC Users without author permission","U of I Only"],"dc:identifier":["AAI9236461","(UMI)AAI9236461","http://hdl.handle.net/2142/20160"],"dc:language":["eng"],"dc:rights":["Copyright 1992 Foguel, Tuval Shmuel"],"dc:subject":["Mathematics"],"dc:title":["Finite groups with a special 2-generator property, and order of centralizers in finite groups"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:15Z"}