Products of Finite Connected Subgroups

María Pilar Gállego, Peter Hauck, Lev S. Kazarin, Ana Martínez-Pastor and María Dolores Pérez-Ramos
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María Pilar Gállego: Departamento de Matemáticas, Universidad de Zaragoza, Edificio Matemáticas, Ciudad Universitaria, 50009 Zaragoza, Spain
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

Mathematics, 2020, vol. 8, issue 9, 1-8

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.

Keywords: finite groups; products of subgroups; two-generated subgroups; L-connection; fitting classes; fitting series; formations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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