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\begin{document}$$\mathfrak{F}$$\end{document} denote a class of groups. A maximal subgroup M of G is called F\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak{F}$$\end{document}-abnormal provided G/MG ∉ F\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak{F}$$\end{document}. We say that (K, H) is an F\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak{F}$$\end{document}-abnormal pair of G provided K is a maximal F\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak{F}$$\end{document}-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-F\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak{F}$$\end{document}-embedded in G if H either covers or avoids every F\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak{F}$$\end{document}-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-F\documentclass[12pt]{minimal}
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\begin{document}$$\mathfrak{F}$$\end{document}-embedded of subgroups.
