Products of finite connected subgroups

[EN] 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 > is an element of L for all a is an element of A and b is an element of 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. 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 third author also by Project VIP-008 of Yaroslavl P. Demidov State University. Gallego, MP.; Hauck, P.; Kazarin, LS.; Martínez-Pastor, A.; Pérez-Ramos, MD. (2020). Products of finite connected subgroups. Mathematics. 8(9):1-8. https://doi.org/10.3390/math8091498 Hauck, P., Kazarin, L. S., Martínez-Pastor, A., & Pérez-Ramos, M. D. (2020). Thompson-like characterization of solubility for products of finite groups. Annali di Matematica Pura ed Applicata (1923 -), 200(1), 337-362. doi:10.1007/s10231-020-00998-z Thompson, J. G. (1968). Nonsolvable finite groups all of whose local subgroups are solvable. Bulletin of the American Mathematical Society, 74(3), 383-437. doi:10.1090/s0002-9904-1968-11953-6 Flavell, P. (1995). Finite groups in which every two elements generate a soluble subgroup. Inventiones Mathematicae, 121(1), 279-285. doi:10.1007/bf01884299 Guralnick, R., Kunyavskiĭ, B., Plotkin, E., & Shalev, A. (2006). Thompson-like characterizations of the solvable radical. Journal of Algebra, 300(1), 363-375. doi:10.1016/j.jalgebra.2006.03.001 CAROCCA, A. (2000). SOLVABILITY OF FACTORIZED FINITE GROUPS. Glasgow Mathematical Journal, 42(2), 271-274. doi:10.1017/s0017089500020139 Carocca, A. (1996). A note on the product of F-subgroups in a finite group. Proceedings of the Edinburgh Mathematical Society, 39(1), 37-42. doi:10.1017/s0013091500022756 Ballester-Bolinches, A., & Pedr