Equations over groupsTools Eljamel, Noha (2018) Equations over groups. PhD thesis, University of Nottingham.
AbstractIf G is a none trivial group and t is an element distinct from G then r(t) =g_1 t^l_1 …......g_kt^l_k= 1 , k ≥1, g_i in G\{1}, l_i in Z\{0} is said to be an equation over G. There has been much research aimed to investigate solvability of such equations over groups and these researches adopted two main approaches. The first considers the properties of G. The other direction which we are following here is to put restrictions on r(t). The results obtained in this direction was of length restriction at first. More resent more results have been obtained and the concept of isolated t-exponent has been introduced and used to study a generalized form of equations of unlimited length. In this study we investigate r(t) which has the generalized form w_1t^l_1w_2t^l_2w_3t^l_3w_4t^l_4 where w_i = g_{i,1} t^m_{i,1} .....t^m_{i,k_i-1}g_{i,k_i}. In Chapter 1 we introduce the concept of equations over groups and we give some of the known results and the geometric method of proof is explained. We also state our main theorem and the main lemma which will be proved in the following chapters and some technical lemmas are proved. In Chapter 2, Chapter 3, and Chapter 4 the Cases I, II, III are discussed and the distribution is shown. In Chapter 5 the main lemma is proved and the proof of the main theorem is completed.
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