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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 generalize the result presented in the book of J. Meldrum  also the results of A. Woryna . 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. We strengthen the results from the author [17, 19] 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. We generalise the results of Meldrum J.  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 A does not have to act faithfully). The commutator of such a group, its minimal generating set and the center of such products has been investigated here. The minimal generating sets for new class of wreath-cyclic geometrical groups and for the commutator of the wreath product are found.
- Laurent Bartholdi, Rostislav I Grigorchuk, and Zoran Suni. Branch groups. In Handbook of algebra, volume 3, Elsevier, ˇ 2003, 989-1112.
- Agnieszka Bier and Vitaliy Sushchansky. Kaluzhnin's representations of sylow p-subgroups of automorphism groups of p-adic rooted trees. Algebra Discr. Math., 19 (2015), 19-38.
- Ievgen V Bondarenko. Finite generation of iterated wreath products. Arch. Math., 95 (4) (2010), 301-308.
- A. Woryna, The rank and generating set for iterated wreath products of cyclic groups, Commun. Algebra, 39 (7) (2011), 2622-2631.
- John Dixon and Brian Mortimer. Permutation groups, Springer Science & Business Media, Volume 163, 1996.
- Andrea Lucchini. Generating wreath products and their augmentation ideals. Rend. Semin. Mat. Univ. Padova, 98 (1997), 67-87.
- Yu V Dmitruk and VI Sushchanskii. Structure of sylow 2-subgroups of the alternating groups and normalizers of sylow subgroups in the symmetric and alternating groups. Ukr. Math. J., 33 (1981), 235-241.
- I. Martin Isaacs. Commutators and the commutator subgroup. Amer. Math. Mon., 9 (1977), 720-722.
- Leo Kaloujnine. Sur les p-groupes de sylow du groupe sym ´etrique du degr ´e pm. C. R. Acad. Sci., 221 (1945), 222-224.
- Yaroslav Lavrenyuk. On the finite state automorphism group of a rooted tree. Algebra Discr. Math., 2002 (2002), 79-87.
- John DP Meldrum. Wreath products of groups and semigroups, volume 74. CRC Press, 1995.
- Alexey Muranov. Finitely generated infinite simple groups of infinite commutator width. Int. J. Algebra Comput., 17 (2007), 607-659.
- Volodymyr Nekrashevych. Self-similar groups, Mathematical Surveys and Monographs, Amer. Math. Soc., New York, Volume 117, 2005.
- Nikolay Nikolov. On the commutator width of perfect groups. Bull. Lond. Math. Soc., 36 (2004), 30-36.
- Vladimir Sharko. Smooth and topological equivalence of functions on surfaces. Ukr. Math. J., 55 (2003), 832-846.
- Ruslan Skuratovskii. Corepresentation of a sylow p-subgroup of a group s n. Cybern. Syst. Anal., 45(1) (2009), 25-37.
- Ruslan Skuratovskii. Minimal generating sets for wreath products of cyclic groups, groups of automorphisms of Ribe graph and fundumental groups of some Morse functions orbits. In Algebra, Topology and Analysis (Summer School), (2016), 121-123.
- Ruslan Skuratovskii. Minimal Generating Set and a Structure of the Wreath Product of Groups, and the Fundamental Group of the Orbit Morse Function. arXiv:1901.00061 [math.GR], (2019), 1-14.
- Ruslan Skuratovskii. Minimal generating sets of cyclic groups wreath product (in russian). In International Conference, Mal'tsev meetting, (2018), 118.
- Vitaly Ivanovich Sushchansky. Normal structure of the isometric metric group spaces of p-adic integers. Algebraic structures and their application. Kiev, (1988), 113-121.
- Sergiy Maksymenko. Deformations of functions on surfaces by isotopic to the identity diffeomorphisms. arXiv:1311.3347 [math.GT], (2013).
- James Wiegold. Growth sequences of finite groups. J. Aust. Math. Soc., 17(2)(1974), 133-141.