On generalised subnormal subgroups of finite groups

All groups considered are finite. Let F be a formation. A subgroup H of a group G is called K-F-subnormal in G if there exists a chain of subgroups H = H-0 subset of H-1 subset of ... subset of H-n = G, with Hi-1 normal in H-i or H-i/Core(Hi) (Hi-1) is an element of F for every 1 <= i <= n. If F = N, the formation of all nilpotent groups, the K-N-subnormal subgroups of a group G are exactly the subnormal subgroups of G. The aim of this paper is to prove the following theorem: if F is a subgroup-closed saturated lattice formation, then a subgroup H of a group G is K-F-subnormal in G if and only if H is K-F-subnormal in < H, x > for all x is an element of G. Some earlier results are consequence of this theorem.