On permutable subgroups of finite groups II

Let beta be a complete set of Sylow subgroups of a finite group G, that is, beta contains exactly one and only one Sylow p-subgroup of G for each prime p. A subgroup H of a finite group G is said to be Beta-permutable if H permutes with every member of Beta. The purpose of this article is to study the influence of Beta-permutability of all maximal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of a finite group G on the structure of G. Our results improve and extend the main results of Asaad (1998), Asaad and Heliel (2003), Asaad et al. (1991), Li et al. (2003), Ramadan (1992), and Srinivasan (1980).