Thompson-like characterization of solubility for products of finite groups

[EN] A remarkable result of Thompson states that a finite group is soluble if and only if all its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups G = AB with subgroups A, B such that is soluble for all a is an element of A and b is an element of B. In this case, the group G is said to be an S-connected product of the subgroups A and B for the class S of all finite soluble groups. Our Main Theorem states that G = AB is S-connected if and only if [A, B] is soluble. In the course of the proof, we derive a result about independent primes regarding the soluble graph of almost simple groups that might be interesting in its own right.Research supported by Proyectos PROMETEO/2017/057 from the Generalitat Valenciana (Valencian Community, Spain) and PGC2018-096872-B-I00 from the Ministerio de Ciencia, Innovacion y Universidades, Spain, and FEDER, European Union; and second author also by Project VIP-008 of Yaroslavl P. Demidov State University.Hauck, P.; Kazarin, LS.; Martínez-Pastor, A.; Pérez-Ramos, MD. (2021). Thompson-like characterization of solubility for products of finite groups. Annali di Matematica Pura ed Applicata (1923 -). 200(1):337-362. https://doi.org/10.1007/s10231-020-00998-z337362200