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Irreducible bases and subgroups of a wreath product in applying to diffeomorphism groups acting on the Möbius band. (English) Zbl 07383951

Summary: We generalize the results presented in the book of Meldrum (Wreath products of groups and semigroups, vol 74, CRC Press, Boca Raton, 1995) about commutator subgroup of wreath products since, as well as considering regular wreath products, we consider those which are not regular (in the sense that the active group \(\mathcal{A}\) does not have to act faithfully). The commutator of such a group, its minimal generating set and the centre of such products has been investigated here. The quotient group of the restricted and unrestricted wreath product by its commutator is found. The generic sets of commutator of wreath product were investigated. The structure of wreath product with non-faithful group action is investigated. Given a permutational wreath product sequence of cyclic groups, we investigate its minimal generating set, the minimal generating set for its commutator and some properties of its commutator subgroup. We strengthen the results from the author (Skuratovskii in Algebra, topology and analysis (summer school), pp 121-123, 2016; International scientific conference. Algebraic and geometric methods of analysis, 2018; The commutator subgroup of Sylow 2-subgroups of alternating group, commutator width of wreath product, arXiv:1903.08765; Minimal generating sets of cyclic groups wreath product (in russian), vol 118, 2018) and construct the minimal generating set for the wreath product of both finite and infinite cyclic groups, in addition to the direct product of such groups. The fundamental group of orbits of a Morse function \(f:M\rightarrow\mathbb{R}\) defined upon a Möbius band \(M\) with respect to the right action of the group of diffeomorphisms \(\mathcal{D}(M)\) has been investigated. In particular, we describe the precise algebraic structure of the group \(\pi_1 O(f)\). A minimal set of generators for the group of orbits of the functions \(\pi_1(O_f,f)\) arising under the action of the diffeomorphisms group stabilising the function \(f\) and stabilising \(\partial M\) have been found. The Morse function \(f\) has critical sets with one saddle point. We consider a new class of wreath-cyclic geometrical groups. The minimal generating set for this group and for the commutator of the group are found.

MSC:

20B27 Infinite automorphism groups
20E08 Groups acting on trees
20B22 Multiply transitive infinite groups
20B35 Subgroups of symmetric groups
20F65 Geometric group theory
20B07 General theory for infinite permutation groups
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[1] Bartholdi, L., Grigorchuk, R.I., Šuni, Z.: Branch groups. In: Handbook of Algebra, vol. 3, pp. 989-1112. Elsevier, North Holland, Amsterdam (2003) · Zbl 1140.20306
[2] Bondarenko, IV, Finite generation of iterated wreath products, Arch. Math., 95, 4, 301-308 (2010) · Zbl 1206.20034
[3] Dixon, JD; Mortimer, B., Permutation Groups (1996), Berlin: Springer, Berlin
[4] Dmitruk, YV; Sushchanskii, VI, Structure of sylow 2-subgroups of the alternating groups and normalizers of sylow subgroups in the symmetric and alternating groups, Ukrain. Math. J., 33, 3, 235-241 (1981) · Zbl 0474.20002
[5] Humphries, S.P.: Generators for the mapping class group. In: Topology of Low-Dimensional Manifolds, pp. 44-47. Springer, Berlin, Heidelberg (1979). ISBN 978-3-540-09506-4
[6] Isaacs, I., Commutators and the commutator subgroup, Am. Math. Mon., 84, 9, 720-722 (1977) · Zbl 0378.20029
[7] Kaloujnine, L., Sur les \(p\)-groupes de sylow du groupe symétrique du degré \(p^m\), C. R. l’Acad. sci., 221, 222-224 (1945) · Zbl 0061.03302
[8] Lavrenyuk, Y., On the finite state automorphism group of a rooted tree, Algebra Discrete Math., 1, 79-87 (2002) · Zbl 1037.20035
[9] Lucchini, A., Generating wreath products and their augmentation ideals, Rendiconti del Seminario Matematico della Università di Padova, 98, 67-87 (1997) · Zbl 0898.20018
[10] Maksymenko, S.: Deformations of functions on surfaces by isotopic to the identity diffeomorphisms. arXiv preprint arXiv:1311.3347 (2013) · Zbl 1468.57031
[11] Meldrum, JDP, Wreath Products of Groups and Semigroups (1995), Boca Raton: CRC Press, Boca Raton · Zbl 0833.20001
[12] Muranov, A., Finitely generated infinite simple groups of infinite commutator width, Int. J. Algebra Comput., 17, 3, 607-659 (2007) · Zbl 1141.20022
[13] Nekrashevych, V., Self-Similar Groups (2005), Providence: American Mathematical Society, Providence · Zbl 1087.20032
[14] Nikolov, N., On the commutator width of perfect groups, Bull. Lond. Math. Soc., 36, 1, 30-36 (2004) · Zbl 1048.20013
[15] Sharko, V., Smooth and topological equivalence of functions on surfaces, Ukrain. Math. J., 55, 5, 832-846 (2003) · Zbl 1039.58036
[16] Skuratovskii, R., Corepresentation of a sylow p-subgroup of a group s n, Cybern. Syst. Anal., 45, 1, 25-37 (2009) · Zbl 1189.20007
[17] Skuratovskii, R.: Minimal generating sets for wreath products of cyclic groups, groups of automorphisms of ribe graph and fundamental groups of some Morse functions orbits. In: Algebra, Topology and Analysis (11-th Summer School 1-14 August), pp. 121-122 (2016) (in russian). http://www.imath.kiev.ua/ topology/ata11
[18] Skuratovskii, R.: The commutator and centralizer description of sylow 2-subgroups of alternating and symmetric groups. arXiv preprint arXiv:1712.01401 (2017) · Zbl 1424.20022
[19] Skuratovskii, R.: Minimal generating sets of cyclic groups wreath product. In: International Conference, Mal’tsev Meeting, p. 118 (2018) (in russian)
[20] Skuratovskii, R., The derived subgroups of sylow 2-subgroups of the alternating group and commutator width of wreath product of groups, Mathematics, 8, 4, 1-19 (2020)
[21] Skuratovskii, R.V.: The commutator subgroup of Sylow 2-subgroups of alternating group, commutator width of wreath product. arXiv:1903.08765 · Zbl 1475.20033
[22] Sushchansky, V.I.: Normal structure of the isometric metric group spaces of p-adic integers. In: Algebraic Structures and Their Application, vol. 17, no. 2, pp. 113-121. Kiev (1988)
[23] Wiegold, J., Growth sequences of finite groups, J. Aust. Math. Soc., 17, 2, 133-141 (1974) · Zbl 0286.20025
[24] Woryna, A., The rank and generating set for iterated wreath products of cyclic groups, Commun. Algebra, 39, 7, 2622-2631 (2011) · Zbl 1231.20030
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