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Products of Finite Connected Subgroups

Author

Listed:
  • María Pilar Gállego

    (Departamento de Matemáticas, Universidad de Zaragoza, Edificio Matemáticas, Ciudad Universitaria, 50009 Zaragoza, Spain
    M. Pilar Gállego passed away on the 22 May 2019. We had the privilege to work with her and to experience her insight and generosity to share her ideas. We miss her as a collaborator and friend.)

  • Peter Hauck

    (Fachbereich Informatik, Universität Tübingen, Sand 13, 72076 Tübingen, Germany)

  • Lev S. Kazarin

    (Department of Mathematics, Yaroslavl P. Demidov State University, Sovetskaya Str 14, 150014 Yaroslavl, Russia)

  • Ana Martínez-Pastor

    (Instituto Universitario de Matemática Pura y Aplicada IUMPA, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain)

  • María Dolores Pérez-Ramos

    (Departament de Matemàtiques, Universitat de València, C/Doctor Moliner 50, 46100 Burjassot (València), Spain)

Abstract

For a non-empty class of groups L , a finite group G = A B is said to be an L -connected product of the subgroups A and B if 〈 a , b 〉 ∈ L for all a ∈ A and b ∈ B . In a previous paper, we prove that, for such a product, when L = S is the class of finite soluble groups, then [ A , B ] is soluble. This generalizes the theorem of Thompson that states the solubility of finite groups whose two-generated subgroups are soluble. In the present paper, our result is applied to extend to finite groups previous research about finite groups in the soluble universe. In particular, we characterize connected products for relevant classes of groups, among others, the class of metanilpotent groups and the class of groups with nilpotent derived subgroup. Additionally, we give local descriptions of relevant subgroups of finite groups.

Suggested Citation

  • María Pilar Gállego & Peter Hauck & Lev S. Kazarin & Ana Martínez-Pastor & María Dolores Pérez-Ramos, 2020. "Products of Finite Connected Subgroups," Mathematics, MDPI, vol. 8(9), pages 1-8, September.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:9:p:1498-:d:408734
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