Transfinite hypercentral iterated wreath product of integral domains

Riccardo Aragona, Norberto Gavioli, Giuseppe Nozzi

Published 2024-11-06, updated 2024-12-20Version 2

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 which characterizes the Lie algebras associated to the Sylow $p$-subgroups of the symmetric group $\mathrm{Sym}(p^n)$. 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 $D^n$.