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 \cite{netreba} 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 $\{{\mathbf{N}}_i{\}}_{i\ge -1}$ starting from the canonical regular abelian subgroup $T$ of $W_n$. Finally, we characterize the regular abelian normal subgroups of ${\mathbf{N}}_0$ that are isomorphic to $D^n$.

This version is a major revision of the previous one, with corrections and new results. Online First. Annali di Matematica (2025)

References in corpus (2)

• Finite $p$-groups of class two with a small multiple holomorph
• The multiple holomorph of centerless groups

Cited by in corpus (1)

• Normality conditions in the Sylow $\mathbf{p}$-subgroup of $\mathrm{Sym}({\mathbf{p}}^{\mathbf{n}})$ and its associated Lie algebra