{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/71252"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/71252","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Rewriting Products of Group Elements","abstract":"Let G be a group and let n be an integer greater than 1. An n-tuple $({\\rm x\\sb1,\\dots, x\\sb{n}})$ of elements of G is said to be rewriteable if there is a nontrivial permutation $\\sigma$ in Sym(n) such that ${\\rm x\\sb1\\dots x\\sb{n} = x\\sb{\\sigma (1)}\\dots x\\sb{\\sigma (n)}}.$ A subset $\\{ {\\rm x\\sb1,\\dots, x\\sb{n}}\\}$ of n elements of G is said to be rewriteable if there exist distinct permutations $\\sigma$ and $\\tau$ in Sym(n) such that ${\\rm x\\sb{\\sigma (1)}\\dots x\\sb{\\sigma (n)} = x\\sb{\\tau (1)}\\dots x\\sb{\\tau (n)}}.$ The group G is totally n-rewriteable (where n $&gt;$ 1), or has the property P $\\sb{\\rm n},$ if every n-tuple $({\\rm x\\sb1,\\dots, x\\sb{n}})$ of n elements of G is rewriteable. G is n-rewriteable, or has the property Q $\\sb{\\rm n},$ if every subset $\\{ {\\rm x\\sb1,\\dots, x\\sb{n}}\\}$ of n elements of G is rewriteable.","abstract_html":"Let G be a group and let n be an integer greater than 1. An n-tuple $({\\rm x\\sb1,\\dots, x\\sb{n}})$ of elements of G is said to be rewriteable if there is a nontrivial permutation <span class=\"etd-inline-math\">&sigma;</span> in Sym(n) such that <span class=\"etd-inline-math\">{\\rm x\\sb1\\dots x\\sb{n} = x\\sb{&sigma; (1)}\\dots x\\sb{&sigma; (n)}}.</span> A subset $\\{ {\\rm x\\sb1,\\dots, x\\sb{n}}\\}$ of n elements of G is said to be rewriteable if there exist distinct permutations <span class=\"etd-inline-math\">&sigma;</span> and $\\tau$ in Sym(n) such that <span class=\"etd-inline-math\">{\\rm x\\sb{&sigma; (1)}\\dots x\\sb{&sigma; (n)} = x\\sb{\\tau (1)}\\dots x\\sb{\\tau (n)}}.</span> The group G is totally n-rewriteable (where n $&amp;gt;$ 1), or has the property P $\\sb{\\rm n},$ if every n-tuple $({\\rm x\\sb1,\\dots, x\\sb{n}})$ of n elements of G is rewriteable. G is n-rewriteable, or has the property Q $\\sb{\\rm n},$ if every subset $\\{ {\\rm x\\sb1,\\dots, x\\sb{n}}\\}$ of n elements of G is rewriteable.","abstract_has_math":true,"creators":["Blyth, Russell David"],"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":1987,"date_issued":"1987","date_published":"1987","updated_at":"2026-07-22T22:26:04Z","subjects":["Mathematics"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["(UMI)AAI8721590"],"render_values":[{"text":"(UMI)AAI8721590","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/71252","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Blyth, Russell David"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["1987","2014-12-16T06:18:18Z","10000-01-01"]},{"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":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/71252","(UMI)AAI8721590"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Let G be a group and let n be an integer greater than 1. An n-tuple $({\\rm x\\sb1,\\dots, x\\sb{n}})$ of elements of G is said to be rewriteable if there is a nontrivial permutation $\\sigma$ in Sym(n) such that ${\\rm x\\sb1\\dots x\\sb{n} = x\\sb{\\sigma (1)}\\dots x\\sb{\\sigma (n)}}.$ A subset $\\{ {\\rm x\\sb1,\\dots, x\\sb{n}}\\}$ of n elements of G is said to be rewriteable if there exist distinct permutations $\\sigma$ and $\\tau$ in Sym(n) such that ${\\rm x\\sb{\\sigma (1)}\\dots x\\sb{\\sigma (n)} = x\\sb{\\tau (1)}\\dots x\\sb{\\tau (n)}}.$ The group G is totally n-rewriteable (where n $&gt;$ 1), or has the property P $\\sb{\\rm n},$ if every n-tuple $({\\rm x\\sb1,\\dots, x\\sb{n}})$ of n elements of G is rewriteable. G is n-rewriteable, or has the property Q $\\sb{\\rm n},$ if every subset $\\{ {\\rm x\\sb1,\\dots, x\\sb{n}}\\}$ of n elements of G is rewriteable.","Every subgroup and factor group of a totally n-rewriteable (respectively, n-rewriteable) group is totally n-rewriteable (respectively, n-rewriteable), and the classes of totally 2-rewriteable groups P $\\sb2$ and 2-rewriteable groups Q $\\sb2$ concide with the class A of abelian groups. Since P $\\sb2 \\subseteq$ P $\\sb3 \\subseteq\\dots$ and Q $\\sb2 \\subseteq$ Q $\\sb3 \\subseteq\\dots,$ the properties P $\\sb{\\rm n}$ and Q $\\sb{\\rm n}$ for n = 3,4,$\\dots$ can be thought of as successively weaker forms of commutativity. Define(UNFORMATTED TABLE OR EQUATION FOLLOWS)$${\\bf P}=\\bigcup\\sbsp{\\rm n=2}{\\infty} {\\bf P}\\sb{\\rm n}\\quad{\\rm and}\\quad {\\bf Q}=\\bigcup\\sbsp{\\rm n=2}{\\infty} {\\bf Q}\\sb{\\rm n};$$(TABLE/EQUATION ENDS) P and Q are the classes of totally rewriteable and rewriteable groups respectively.","Previous authors have investigated the total rewriting properties; the current author studies mainly the rewriting properties. The principal result is a complete characterization of the rewriteable groups: a group is rewriteable if and only if it is finite-by-abelian-by-finite, that is, if it has a normal subgroup N such that $\\vert$G:N$\\vert$ and ${\\rm N}\\sp\\prime$ are finite. The second main result is concerned with semisimple groups, that is, groups which possess no non-trivial normal abelian subgroups. For each n $\\ge$ 3, there is a bound ${\\rm J\\sb{n}}$ on the order of a semisimple Q $\\sb{\\rm n}$-group. The final main result shows that every Q $\\sb4$-group is solvable. Elementary FC-group results, information about the structure of groups of Lie type, and the classifications of the finite simple groups and of the minimal simple groups are used to prove these results.","Made available in DSpace on 2014-12-16T06:18:18Z (GMT). No. of bitstreams: 1 8721590.pdf: 3432045 bytes, checksum: 2bfd39bea81f24a8f0d5fc0076da06b3 (MD5) Previous issue date: 1987","Embargo set by: Seth Robbins for item 71418 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","157 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987."]},{"key":"dc:title","label":"Title","values":["Rewriting Products of Group Elements"]}]}],"canonical_facts":{"dc:creator":["Blyth, Russell David"],"dc:date":["1987","2014-12-16T06:18:18Z","10000-01-01"],"dc:description":["Let G be a group and let n be an integer greater than 1. An n-tuple $({\\rm x\\sb1,\\dots, x\\sb{n}})$ of elements of G is said to be rewriteable if there is a nontrivial permutation $\\sigma$ in Sym(n) such that ${\\rm x\\sb1\\dots x\\sb{n} = x\\sb{\\sigma (1)}\\dots x\\sb{\\sigma (n)}}.$ A subset $\\{ {\\rm x\\sb1,\\dots, x\\sb{n}}\\}$ of n elements of G is said to be rewriteable if there exist distinct permutations $\\sigma$ and $\\tau$ in Sym(n) such that ${\\rm x\\sb{\\sigma (1)}\\dots x\\sb{\\sigma (n)} = x\\sb{\\tau (1)}\\dots x\\sb{\\tau (n)}}.$ The group G is totally n-rewriteable (where n $&gt;$ 1), or has the property P $\\sb{\\rm n},$ if every n-tuple $({\\rm x\\sb1,\\dots, x\\sb{n}})$ of n elements of G is rewriteable. G is n-rewriteable, or has the property Q $\\sb{\\rm n},$ if every subset $\\{ {\\rm x\\sb1,\\dots, x\\sb{n}}\\}$ of n elements of G is rewriteable.","Every subgroup and factor group of a totally n-rewriteable (respectively, n-rewriteable) group is totally n-rewriteable (respectively, n-rewriteable), and the classes of totally 2-rewriteable groups P $\\sb2$ and 2-rewriteable groups Q $\\sb2$ concide with the class A of abelian groups. Since P $\\sb2 \\subseteq$ P $\\sb3 \\subseteq\\dots$ and Q $\\sb2 \\subseteq$ Q $\\sb3 \\subseteq\\dots,$ the properties P $\\sb{\\rm n}$ and Q $\\sb{\\rm n}$ for n = 3,4,$\\dots$ can be thought of as successively weaker forms of commutativity. Define(UNFORMATTED TABLE OR EQUATION FOLLOWS)$${\\bf P}=\\bigcup\\sbsp{\\rm n=2}{\\infty} {\\bf P}\\sb{\\rm n}\\quad{\\rm and}\\quad {\\bf Q}=\\bigcup\\sbsp{\\rm n=2}{\\infty} {\\bf Q}\\sb{\\rm n};$$(TABLE/EQUATION ENDS) P and Q are the classes of totally rewriteable and rewriteable groups respectively.","Previous authors have investigated the total rewriting properties; the current author studies mainly the rewriting properties. The principal result is a complete characterization of the rewriteable groups: a group is rewriteable if and only if it is finite-by-abelian-by-finite, that is, if it has a normal subgroup N such that $\\vert$G:N$\\vert$ and ${\\rm N}\\sp\\prime$ are finite. The second main result is concerned with semisimple groups, that is, groups which possess no non-trivial normal abelian subgroups. For each n $\\ge$ 3, there is a bound ${\\rm J\\sb{n}}$ on the order of a semisimple Q $\\sb{\\rm n}$-group. The final main result shows that every Q $\\sb4$-group is solvable. Elementary FC-group results, information about the structure of groups of Lie type, and the classifications of the finite simple groups and of the minimal simple groups are used to prove these results.","Made available in DSpace on 2014-12-16T06:18:18Z (GMT). No. of bitstreams: 1 8721590.pdf: 3432045 bytes, checksum: 2bfd39bea81f24a8f0d5fc0076da06b3 (MD5) Previous issue date: 1987","Embargo set by: Seth Robbins for item 71418 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","157 p.","Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987."],"dc:identifier":["http://hdl.handle.net/2142/71252","(UMI)AAI8721590"],"dc:subject":["Mathematics"],"dc:title":["Rewriting Products of Group Elements"],"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":"2026-07-22T22:26:04Z"}