Transfinite hypercentral iterated wreath product of integral domains

Abstract 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 $${{\,\textrm{Sym}\,}}(p^n)$$ . As an application, we explore the normalizer chain $$\lbrace \textbf{N}_{i}\rbrace _{i\ge -1}$$ starting from the canonical regular abelian subgroup T of $$W_n$$ . Finally, we characterize the regular abelian normal subgroups of $$\textbf{N}_0$$ that are isomorphic to $$D^n$$ .