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Title:Rewriting Products of Group Elements
Author(s):Blyth, Russell David
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
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 $>$ 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.
Issue Date:1987
Description:157 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987.
Other Identifier(s):(UMI)AAI8721590
Date Available in IDEALS:2014-12-16
Date Deposited:1987

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