Torsion-by-nilpotent groups

In this thesis, the article written by Tarek Rouabhi and Nadir Trabelsi titled "A note on Torsion-by-Nilpotent Groups" was studied. In the aforementioned article, groups with nilpotent extension of torsion groups were investigated. First of all, to prove a few results in this thesis, we considered the article titled "Soluble groups with many 2-generator torsion-by-nilpotent subgroups" written by Nadir Trabelsi. In the fourth chapter of this thesis, we examined class (X,∞), when X is class of torsion-by-nilpotent groups and proof of the theorem " Let G be a finitely generated soluble group in the class (TN,∞). Then G is torsion-by-nilpotent" was given in detail. When the results of [3], [10] and this theorem were combined, some results regarding the class (TN,∞) were obtained. These results obtained in the fifth chapter were extended to the (TN,∞)^* class in the article by Tarek Rouabhi and Nadir Trabelsi. Thus, the main result of our study, the proof of the theorem "A finitely generated soluble-by-finite group in the class (TN,∞)^* is torsion-by-nilpotent" was shown in detail. When the results of [2] and this theorem were combined in our study, the following results were obtained: "Let k be a positive integer (i) If G is a finitely generated soluble-by-finite group in the class (TN_k,∞)^*, then there exists an integer c=c(k), depending only on k, such that G belongs to TN_c (ii) A finitely generated metabelian-by-finite group is in the class (TN_k,∞)^* if, and only if, it belongs to TN_(k+1)". In the fifth chapter, the proof of these results was given in detail.