On generalised subnormal subgroups of finite groups

Let F be a formation of finite groups. A subgroup M of a finite group G is said to be F-normal in G if Core(G) (M) belongs to F. A subgroup U of a finite group G is called a K-F-subnormal subgroup of G if either U = G or there exist subgroups U = U-0 U-1 <= U-1 <= ... <= U-n = G such that Ui - 1 is either normal or F-normal in U-i, for i = 1, 2, ..., n. The K-F-subnormality could be regarded as the natural extension of the subnormality to formation theory and plays an important role in the structural study of finite groups. The main purpose of this paper is to analyse classes of finite groups whose K-F-subnormal subgroups are exactly the subnormal ones. Some interesting extensions of well-known classes of groups emerge. (C) 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim