On locally normal Fitting classes of finite soluble groups (Q1868921)

A generalization of the concept of normal Fitting classes introduced by Blessenohl and Gaschütz arises in the following way: Let \(\mathcal X\) and \(\mathcal F\) be non-trivial Fitting classes of finite soluble groups such that \({\mathcal X}\subseteq{\mathcal F}\). Then \(\mathcal X\) is said to be normal in \(\mathcal F\) if an \(\mathcal X\)-injector of \(G\) is a normal subgroup of \(G\) for every \(G\in{\mathcal F}\). In the paper the author restricts attention to the case in which both classes, \(\mathcal X\) and \(\mathcal F\) are subgroup-closed Fitting classes and therefore they are locally defined formations. This yields a good behaviour with respect to the corresponding canonical local definitions. First the author establishes the existence of a unique maximal subgroup-closed Fitting class in which a given subgroup-closed Fitting class \(\mathcal X\) is normal. Moreover, the class \(\mathcal X\) is determined uniquely by this class, and in many cases, such as when \(\mathcal X\) is of bounded nilpotent length there is an algorithm to describe it. Furthermore, it is proved that the collection of all subgroup-closed Fitting classes in which \(\mathcal X\) is normal forms a complete, distributive and atomic lattice, whose atoms can be described explicitly. The dual problem, that is, the existence of a unique minimal subgroup-closed Fitting class being normal in a given subgroup-closed Fitting class \(\mathcal F\) is still an open question. However the author shows that if \(\mathcal F\) is a subgroup-closed Fitting class such that a smallest \(\mathcal F\)-normal subgroup-closed Fitting class exists, then the collection of all \(\mathcal F\)-normal subgroup-closed Fitting classes forms also a complete and distributive lattice. Moreover it is dual atomic if \(\mathcal F\) is of bounded nilpotent length.