The product of two finite nilpotent groups and its Fitting series (Q1203556)

A well-known theorem of Kegel and Wielandt says that a finite product \(G = AB\) of two nilpotent subgroups \(A\) and \(B\) is soluble; also the Fitting subgroup \(F = F(G)\) of \(G\) satisfies \(F = (A \cap F)(B \cap F)\) as was shown independently by Pennington and the reviewer. But it seems hard to decide whether the solubility length of \(G\) is bounded by a function depending on the nilpotency classes of \(A\) and \(B\). Here it is proved that there is a bound for the Fitting length of \(G\) if \(A\) and \(B\) belong to certain subclasses of nilpotent groups with the nilpotency class of the Sylow \(p\)-subgroups bounded by a function of \(p\) (Theorem 1). In the proof a classical result of Hall and Higman plays a role. It is also shown that the subgroup \(F(F_ 2 \cap A)\) is normal in \(G\), where \(F_ 2/F\) is the Fitting subgroup of \(G/F\) (Theorem 2). The proof uses information on the structure of primitive products of two nilpotent groups.