Transfinite hypercentral iterated wreath product of integral domains

TL;DR

This paper constructs a class of transfinite hypercentral iterated wreath products of integral domains, linking them to Lie rings and exploring their algebraic structure and normalizers.

Contribution

It introduces a new class of wreath products based on numerical polynomials and establishes a correspondence with Lie rings, extending previous algebraic characterizations.

Findings

Generalizes Lie algebra characterization for wreath products

Identifies conditions for transfinite hypercentrality

Characterizes regular abelian normal subgroups

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 $\lbrace\mathbf{N}_{i}\rbrace_{i\geq -1}$ 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 $D^n$.