Irreducible Bases and Subgroups of a Wreath Product in Applying to Diffeomorphism Groups acting on the M\'obius Band
| Autor: | Skuratovskii, Ruslan Vyacheslavovich, Williams, Aled |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Rendiconti del Circolo Matematico di Palermo Series 2021. 2, 70(2), 721-739 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s12215-020-00514-5 |
| Popis: | Given a permutational wreath product sequence of cyclic groups we investigate its minimal generating set, minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the result of author \cite{SkVC, SkMal, SkAr} and construct minimal generating set for wreath product of finite and infinite cyclic groups and direct product of such groups. We generalize results of Meldrum about commutator subgroup of wreath product \cite{Meld} because we take in consideration as regular wreath product as well as no regular (where active group $\mathcal{A}$ can acts not faithfully). Also commutator of such group and its minimal generating set. Also center of such products was investigated. Also fundamental group of orbits of a Morse function $f:M\to \mathbb{R}$ defined on a Mebius band $M$ with respect to the right action of the group of diffeomorphisms $\mathcal{D}(M)$ is investigated by us. The paper describes precise algebraic structure of the group $\pi_1 O(f)$. A minimal set of generators for the group of orbits of functions ${{\pi }_{1}}({{O}_{f}},f)$ arising under the action of diffeomorphisms group stabilizing the function $f$ and stabilizing $\partial M$ is found. The the Morse function $f$ has critical sets with one saddle point. The quotient group of restricted wreath products by its commutator was found. The generic sets of commutator of wreath product were investigated. Minimal generating set for this group and for commutator of group are found. This paper after previous Arxiv versions from 2019 \cite{SkArM, SkArM3} with previous title "Minimal generating set and structure of wreath product of groups with non-faithful action, comutator subgroup of wreath product and the fundamental group of orbit of Morse function $\pi_1 O(f)$" was published \cite{SkRendi}. Comment: 2 figures, 3 algebraic international conferences |
| Databáze: | arXiv |
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