Abstract
For a non-empty class of groups L, a finite group G=AB 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.
Highlights
Introduction and Main ResultsAll of the groups considered in this paper are assumed to be finite
We extend previous research on the influence of two-generated subgroups on the structure of groups, in connection with the study of products of subgroups
Let the finite group G = AB be the product of subgroups A and B
Summary
María Pilar Gállego 1,† , Peter Hauck 2 , Lev S. Departament de Matemàtiques, Universitat de València, C/Doctor Moliner 50, 46100 Burjassot (València), 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. Received: 5 August 2020; Accepted: 1 September 2020; Published: 4 September 2020
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.