Produkte Abelscher Gruppen und ihre Fittinggruppe. (Products of Abelian groups and their Fitting group) (Q581541)

A famous theorem of \textit{N. Itô} [Math. Z. 62, 400-401 (1955; Zbl 0064.25203)] states that every group which is the product of two abelian subgroups is metabelian. However the size of the commutator subgroup of such a group is not in general closely related to the size of the factors. In this paper the author shows that the Fitting subgroup \(F(G)\) of a product \(G=AB\) of two abelian groups behaves much better. In fact he proves the following theorem. Let the group \(G=AB\) be the product of two abelian subgroups \(A\) and \(B\). (1) If B is finite, then the Fitting factor group \(G/F(G)\) is finite with order at most \(|B|\). (2) If \(B\) is finitely generated, then \(G/F(G)\) is finitely generated and its torsion-free rank is less or equal than the torsion-free rank of \(B\). -- Examples show that the nilpotency class of the Fitting subgroup of such a group cannot be bounded.