Transfinite hypercentral iterated wreath product of integral domains

2024/11/06 by Aragona, Riccardo, Gavioli, Norberto, Nozzi, Giuseppe
#05A17 #20B35 #20E15 #20E22 #20F19 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2411.03846

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 \citenetreba which characterizes the Lie algebras associated to the Sylow \(p\)-subgroups of the symmetric group \(\Sym(pⁿ)\). As an application, we explore the normalizer chain \lbraceN_i\rbrace_{i≥ -1} starting from the canonical regular abelian subgroup T of W_n. Finally, we characterize the regular abelian normal subgroups of N₀ that are isomorphic to Dⁿ.

Related

• On the normalizer of an iterated wreath product
• Normal subgroups of iterated wreath products of symmetric groups and alternating with symmetric groups
• Natural transformations between induction and restriction on iterated wreath product of symmetric group of order 2
• Stable Centres I: Wreath Products
• Induction/Restriction Bialgebras for Restricted Wreath Products