Two Different Approaches to the Study of Abelian Regular Subgroups of the Symmetric Group

Let \(n \ge 2\) and \(V\) be a vector space over a finite field of dimension \(n\). This thesis is devoted to the study of elementary abelian regular subgroups of the symmetric group \(\Sym(V)\). After recalling the necessary background and summarizing previous results in the literature, we investigate the \(p\)-Sylow subgroup \(W_n\) of the symmetric group on \(p^n\) elements. We introduce a weighted degree order on its elements, compute its central series, and study the Lie algebra associated with its lower central series. Moreover we study the normal subgroups of \(W_n\) and the normalizer chain originating from the canonical elementary abelian regular subgroup of \(W_n\). We extend this analysis to characteristic zero by constructing an analogue of \(W_n\) as an iterated wreath product over an integral domain, proving that it is transfinite hypercentral and explicitly describing its ascending central series and normal subgroups. In the final part of the thesis, we study abelian regular subgroups of affine groups over free modules by exploiting their correspondence with bi-brace structures. Under suitable assumptions, we reduce the classification problem to the study of bilinear forms, and we obtain classification results over finite fields of odd characteristic and over the ring of \(p\)-adic integers, both in the torsion-free and torsion cases.