Finite Groups with Systems of Σ-$$\mathfrak{F}$$-Embedded Subgroups

Let $$\mathfrak{F}$$ denote a class of groups. A maximal subgroup M of G is called $$\mathfrak{F}$$ -abnormal provided G/MG ∉ $$\mathfrak{F}$$ . We say that (K, H) is an $$\mathfrak{F}$$ -abnormal pair of G provided K is a maximal $$\mathfrak{F}$$ -abnormal subgroup of H. Let Σ = {G0 ≤ G1 ≤ G2 ≤ … ≤ Gn} be a subgroup series of G. A subgroup H of G is said to be Σ- $$\mathfrak{F}$$ -embedded in G if H either covers or avoids every $$\mathfrak{F}$$ -abnormal pair (K, H) such that Gi−1≤ K < H ≤ Gi for some i ∈ {0, 1, …, n}. In this paper, some new characterizations of p-supersoluble and p-soluble are given by discussing the properties of Σ- $$\mathfrak{F}$$ -embedded of subgroups.