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

Abstract

Let $\mathfrak{F}$ denote a class of groups. A maximal subgroup M of G is called $\mathfrak{F}$-abnormal provided G/M_G ∉ $\mathfrak{F}$. 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 ≤ … ≤ G_n} 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 G_i−1≤ K < H ≤ G_i 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.