Normality conditions in the Sylow $\boldsymbol{p}$-subgroup of $\boldsymbol{\mathrm{Sym}(p^n)}$ and its associated Lie algebra
Riccardo Aragona, Norberto Gavioli, Giuseppe Nozzi
TL;DR
The paper studies normality conditions in the Sylow $p$-subgroup $W_n$ of $\mathrm{Sym}(p^n)$ by embedding $W_n$ as an iterated wreath product and relating its central series to a graded Lie algebra $\mathfrak{L}_n$. It proves the upper and lower central series of $W_n$ coincide and describes normal subgroups that intersect tail base subgroups, obtaining a bounded index relative to lower-central terms. A Lie-algebra perspective is developed via a map $\varphi: W_n\to \mathfrak{L}_n$, yielding a bijection between normal saturated subgroups and homogeneous ideals, and enabling parallel central-series results in $\mathfrak{L}_n$. Finally, the growth of the normalizer chain from the canonical abelian subgroup is shown to match an idealizer chain in $\mathfrak{L}_n$, with explicit growth $|N_i/N_{i-1}|=p^{q_{p,i+1}}$ determined by partition counts, connecting to known integer sequences. This provides a cohesive group-Lie algebra framework for normality and normalizer growth in Sylow $p$-subgroups of symmetric groups.
Abstract
In this work, we give a description of the structure of the normal subgroups of a Sylow $p$-subgroup $W_n$ of $\mathrm{Sym}(p^n)$, showing that they contain a term from the lower central series with bounded index. To this end, we explicitly determine the terms of the upper and the lower central series of $W_n$. We provide a similar description of these series in the Lie algebra associated to $W_n$, giving a new proof of the equality of their terms in both the group and the algebra contexts. Finally, we calculate the growth of the normalizer chain starting from an elementary abelian regular subgroup of $W_n$.