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Title:  Rewriting Products of Group Elements 
Author(s):  Blyth, Russell David 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  Let G be a group and let n be an integer greater than 1. An ntuple $({\rm x\sb1,\dots, x\sb{n}})$ 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 nrewriteable (where n $>$ 1), or has the property P $\sb{\rm n},$ if every ntuple $({\rm x\sb1,\dots, x\sb{n}})$ of n elements of G is rewriteable. G is nrewriteable, 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 nrewriteable (respectively, nrewriteable) group is totally nrewriteable (respectively, nrewriteable), and the classes of totally 2rewriteable groups P $\sb2$ and 2rewriteable 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 finitebyabelianbyfinite, 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 nontrivial 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 FCgroup 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. 
Issue Date:  1987 
Type:  Text 
Description:  157 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1987. 
URI:  http://hdl.handle.net/2142/71252 
Other Identifier(s):  (UMI)AAI8721590 
Date Available in IDEALS:  20141216 
Date Deposited:  1987 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois