Fitting subgroups of profinite completions of soluble minimax groups
This paper is concerned with the question of how far a group is determined by the class of all of its finite images. It is known that it is rarely determined up to isomorphism but for some classes of groups, there can be only finitely many isomorphism types of groups of that class with the same collection of finite images. This was proved for finitely generated nilpotent groups by \textit{P. F. Pickel} [Trans. Am. Math. Soc. 160, 327-341 (1971; Zbl 0235.20027)] and for polycyclic-by-finite groups by \textit{F. Grunewald}, \textit{P. Pickel} and \textit{D. Segal} [Ann. Math., II. Ser. 111, 155-195 (1980; Zbl 0431.20033)]. In [Arch. Math. 43, 303- 311 (1984; Zbl 0526.20026)], the author proved that it was also true for finite extensions of torsion-free nilpotent minimax groups G which are divisible by all primes in the set of primes p for which G has a quasi- cyclic section (the spectrum of G). The author here makes a contribution towards improving this theorem by replacing ''torsion-free nilpotent'' by ''soluble'' and so providing a common generalization to all of these results. The main result is proved is the following: Theorem. The Fitting subgroup of the profinite completion of a finite extension of a residually finite soluble minimax group coincides with the profinite completion of the Fitting subgroup. Combined with earlier results of the author, this yields the following contribution to the above-mentioned problem. Theorem. Let C be a class of residually finite soluble-by-finite minimax groups with the same finite images. Then (1) The groups in C have the same spectrum \(\pi\) and Hirsch number; (2) There exists a natural number n such that, for all \(G\in C\) the n-th power, N(G), of the Fitting subgroup of G is torsion-free nilpotent; (3) The class of all \(\pi\) '-localizations of N(G) splits into finitely many isomorphism classes.
finite extensions of torsion-free nilpotent minimax groups
scientific article; zbMATH DE number 3946438