Transfinite hypercentral iterated wreath product of integral domains

Starting with an integral domain D of characteristic 0, we consider a class of iterated wreath product W_n of n copies of D. In order that W_n be transfinite hypercentral, it is necessary to restrict to the case of wreath products defined by way of numerical polynomials. We also associate to each of these groups a Lie ring, providing a correspondence preserving most of the structure. This construction generalizes a result of Sushchansky and Netreba (Algebra Discrete Math 122–132, 2005) which characterizes the Lie algebras associated to the Sylow p-subgroups of the symmetric group Sym(p^n). As an application, we explore the normalizer chain {N_i} starting from the canonical regular abelian subgroup T of W_n. Finally, we characterize the regular abelian normal subgroups of N_0 that are isomorphic to D^n.