On locally normal Fitting classes of finite soluble groups

Let , be non-trivial Fitting classes of finite soluble groups such that is an -injector of G for all . Then is said to be normal in ( -normal). We show that for a subgroup-closed Fitting class the collection of all subgroup-closed Fitting classes in which is normal forms a complete, distributive and atomic lattice. Moreover, is determined uniquely by the unique maximal subgroup-closed Fitting class in which is normal, and in many cases there is an algorithm to describe this class explicitly. Further we show that if is a subgroup-closed Fitting class such that a unique minimal subgroup-closed -normal Fitting class exists, the collection of all subgroup-closed -normal Fitting classes also forms a complete distributive lattice which in addition is dual atomic if is of bounded nilpotent length.