Transfinite hypercentral iterated wreath product of integral domains
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866916975246049280 |
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| author | Aragona, Riccardo Gavioli, Norberto Nozzi, Giuseppe |
| author_facet | Aragona, Riccardo Gavioli, Norberto Nozzi, Giuseppe |
| contents | 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 $D^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_03846 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Transfinite hypercentral iterated wreath product of integral domains Aragona, Riccardo Gavioli, Norberto Nozzi, Giuseppe Group Theory Combinatorics 20E22, 20B35, 20E15, 20F19, 05A17 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 $D^n$. |
| title | Transfinite hypercentral iterated wreath product of integral domains |
| topic | Group Theory Combinatorics 20E22, 20B35, 20E15, 20F19, 05A17 |
| url | https://arxiv.org/abs/2411.03846 |