Finite Groups with Systems of Σ-F-Embedded Subgroups

Let F denote a class of groups. A maximal subgroup M of G is called F-abnormal provided G/M-G is not an element of F. We say that (K, H) is an F-abnormal pair of G provided K is a maximal F-abnormal subgroup of H. Let Sigma = {G(0) <= G(1) <= G(2) <= ... <= G(n)} be a subgroup series of G. A subgroup H of G is said to be Sigma-F-embedded in G if H either covers or avoids every F-abnormal pair (K, H) such that G(i-1)<= K < H <= G(i) for some i is an element of {0, 1, ..., n}. In this paper, some new characterizations of p-supersoluble and p-soluble are given by discussing the properties of Sigma-F-embedded of subgroups.