Irreducible bases and subgroups of a wreath product in applying to diffeomorphism groups acting on the Möbius band


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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2


  1. 1.

    Bartholdi, L., Grigorchuk, R.I., Šuni, Z.: Branch groups. In: Handbook of Algebra, vol. 3, pp. 989–1112. Elsevier, North Holland, Amsterdam (2003)

  2. 2.

    Bondarenko, I.V.: Finite generation of iterated wreath products. Arch. Math. 95(4), 301–308 (2010)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Dixon, J.D., Mortimer, B.: Permutation Groups, vol. 163. Springer, Berlin (1996)

    Google Scholar 

  4. 4.

    Dmitruk, Y.V., Sushchanskii, V.I.: 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)

    Article  Google Scholar 

  5. 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. 6.

    Isaacs, I.: Commutators and the commutator subgroup. Am. Math. Mon. 84(9), 720–722 (1977)

    MathSciNet  Article  Google Scholar 

  7. 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)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Lavrenyuk, Y.: On the finite state automorphism group of a rooted tree. Algebra Discrete Math. 1, 79–87 (2002)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Lucchini, A.: Generating wreath products and their augmentation ideals. Rendiconti del Seminario Matematico della Università di Padova 98, 67–87 (1997)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Maksymenko, S.: Deformations of functions on surfaces by isotopic to the identity diffeomorphisms. arXiv preprint arXiv:1311.3347 (2013)

  11. 11.

    Meldrum, J.D.P.: Wreath Products of Groups and Semigroups, vol. 74. CRC Press, Boca Raton (1995)

    Google Scholar 

  12. 12.

    Muranov, A.: Finitely generated infinite simple groups of infinite commutator width. Int. J. Algebra Comput. 17(03), 607–659 (2007)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Nekrashevych, V.: Self-Similar Groups, vol. 117. American Mathematical Society, Providence (2005)

    Google Scholar 

  14. 14.

    Nikolov, N.: On the commutator width of perfect groups. Bull. Lond. Math. Soc. 36(1), 30–36 (2004)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Sharko, V.: Smooth and topological equivalence of functions on surfaces. Ukrain. Math. J. 55(5), 832–846 (2003)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Skuratovskii, R.: Corepresentation of a sylow p-subgroup of a group s n. Cybern. Syst. Anal. 45(1), 25–37 (2009)

    MathSciNet  Article  Google Scholar 

  17. 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).

  18. 18.

    Skuratovskii, R.: The commutator and centralizer description of sylow 2-subgroups of alternating and symmetric groups. arXiv preprint arXiv:1712.01401 (2017)

  19. 19.

    Skuratovskii, R.: Minimal generating sets of cyclic groups wreath product. In: International Conference, Mal'tsev Meeting, p. 118 (2018) (in russian)

  20. 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)

    Article  Google Scholar 

  21. 21.

    Skuratovskii, R.V.: The commutator subgroup of Sylow 2-subgroups of alternating group, commutator width of wreath product. arXiv:1903.08765

  22. 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. 23.

    Wiegold, J.: Growth sequences of finite groups. J. Aust. Math. Soc. 17(2), 133–141 (1974)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Woryna, A.: The rank and generating set for iterated wreath products of cyclic groups. Commun. Algebra 39(7), 2622–2631 (2011)

    MathSciNet  Article  Google Scholar 

Download references


We are grateful to Antonenko Alexandr for a graphical support and Sergey Maksymenko for Morse function description, also we thanks to Samoilovych I.

Author information



Corresponding author

Correspondence to Ruslan V. Skuratovskii.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Skuratovskii, R.V., Williams, A. Irreducible bases and subgroups of a wreath product in applying to diffeomorphism groups acting on the Möbius band. Rend. Circ. Mat. Palermo, II. Ser (2020).

Download citation


  • Wreath product
  • Minimal generating set of commutator subgroup
  • Center of non regular wreath product
  • Quotient by commutator subgroup of wreath product
  • Semidirect product
  • Fundamental group of orbits of one Morse function

Mathematics Subject Classification

  • 20B27
  • 20E08
  • 20B22
  • 20B35
  • 20F65
  • 20B07