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.

让\（\ mathfrak {F} \）表示一组类别。极大子群中号的ģ称为\（\ mathfrak {F} \） -abnormal提供G / M _ģ ∉ \（\ mathfrak {F} \） 。我们说（K，H）是G的\（\ mathfrak {F} \）-异常对，前提是K是H的最大\（\ mathfrak {F} \）-异常子组。让Σ= { g ^ ₀ ≤ ģ ₁ ≤ ģ ₂ ≤...≤ ģ _Ñ }是一个亚组一系列ģ。一个小组ħ的ģ据说是Σ- \（\ mathfrak {F} \）在-嵌入式ģ如果ħ任一盖或避免每\（\ mathfrak {F} \） -abnormal对（K，H），使得ģ_我-1 ≤ķ<H≤ģ_我一些我∈{0，1，...，ñ }。在本文中，一些新的表征p -supersoluble和p可溶被讨论Σ-的特性给定的\（\ mathfrak {F} \） -嵌入式子组。