Fitting subgroups of profinite completions of soluble minimax groups

Abstract

How far does the class of all finite images of a group determine this group? It is known that finitely generated abelian groups are determined by their finite images up to isomorphism whereas finitely generated nilpotent groups in general are not [4]. At least there are only finitely many nonisomorphic such groups with the same finite images [S]. The same is true for polycyclic-by-finite groups [3] and for finite extensions of torsion-free nilpotent minimax groups which are radicable by its spectrum [7]. It is not known if there is a common generalization to soluble minimax groups the Fitting subgroups of which are radicable by their spectrum. In the way Pickel’s theorem [9] was a first step to the proof of the result on polycyclic groups the Main Theorem of this paper might be helpful on the way of a proof of such a more general result.