2019/08/09 by Peter Hauck, Hauck, P., Л. С. Казарин +5
Engineering · Mathematics · #20D10 #20D40 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1908.03347
openalex publication_date 2019/08/09 · openalex created_date 2019/08/22 · openalex updated_date 2026/07/28
A remarkable result of Thompson states that a finite group is soluble if and only if 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 ⟨ a, b⟩ is soluble for all a ∈ A and b ∈ B. In this case, the group G is said to be an \cal S-connected product of the subgroups A and B for the class \cal S of all finite soluble groups. Our main theorem states that G = AB is \cal S-connected if and only if [A,B] is soluble. In the course of the proof we derive a result of own interest about independent primes regarding the soluble graph of almost simple groups.