Normality conditions in the Sylow p-subgroup of Sym(p^n) and its associated Lie algebra

In this work, we give a description of the structure of the normal subgroups of a Sylow p-subgroup W_n of 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 algebra contexts. Finally, we calculate the growth of the normalizer chain starting from an elementary abelian regular subgroup of W_n.