More Metanilpotent Fitting Classes with Bounded Chief Factor Ranks

Metanilpotent Fitting classes are given consisting of groups with elementary abelian Fitting quotient and fairly wide choices of the nilpotency class of the nilpotent residual. Many of the minimal normal subgroups are non-central. AMS subject classiication 20D10 1 Key sections The aim of this note is to establish further examples of Fitting classes of metanilpotent groups to further the transparence of this family of Fitting classes. Our main interest will be in further classes with groups that have nilpotent residual of bounded nilpotency class, and we will restrict ourselves to Fitting classes F in S p S q. In contrast to the examples given by Menth 7] and Traustason 8] the families given here do not have central socle but the central chief factors will still play an important role. For the description of the Fitting classes we will use a concept rst introduced by Dark 1], the key section. A key section of a Fitting class F consists, for given sets ; of primes, of all sections O (G=O (G)) of groups G 2 F. In our case the only interesting key section is the one consisting of the sections O p (G=O q (G)), and we will denote it by KF. It is comparatively easy to see that there is a one-to-one correspondence of nontrivial key sections KF to nonnilpotent Fitting classes F in S p S q. For a given key section of this family the corresponding Fitting class is achieved by applying the following two operations, beginning with a member of the key section: (a) normal products with p-groups, (b) subdirect products with q-groups. The following two facts are not too diicult to establish: