The following paper has been published: El siguiente artículo ha sido publicado: El següent article ha sigut publicat:
Rex Darl, Arnold D. Feldman, M. D. Pérez-Ramos. Nilpotent length and system permutability. J. Algebra, 589:287-322, 2022.
If C is a class of groups, a C-injector of a finite group G is a subgroup V of G with the property that V ∩ K is a C-maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschütz and B. Hartley states the existence and conjugacy of F-injectors in finite soluble groups for Fitting classes F. We shall show that for groups of nilpotent length at most 4, F-injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.
The aim of this paper is to give answers to some questions concerning intersections of system normalisers and prefrattini subgroups of finite soluble groups raised by the third author, Shemetkov and Vasil’ev in the Kourovka Notebook . Our approach depends on results on regular orbits and it can be also used to extend a result of Mann  concerning intersections of injectors associated to Fitting classes.