Skip to main content
Log in

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

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

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 via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  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. 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)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

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

    MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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)

  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)

    Article  Google Scholar 

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

  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)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

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

Authors and Affiliations

Authors

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

Check for updates. 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 70, 721–739 (2021). https://doi.org/10.1007/s12215-020-00514-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-020-00514-5

Keywords

Mathematics Subject Classification

Navigation