All groups considered are finite. Let F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} be a formation. A subgroup H of a group G is called K-F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document}-subnormal in G if there exists a chain of subgroups H=H0⊆H1⊆⋯⊆Hn=G,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} H=H_{0} \subseteq H_{1} \subseteq \cdots \subseteq H_{n}=G, \end{aligned}$$\end{document}with Hi-1\documentclass[12pt]{minimal}
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\begin{document}$$H_{i-1}$$\end{document} normal in Hi\documentclass[12pt]{minimal}
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\begin{document}$$H_{i}$$\end{document} or Hi/CoreHi(Hi-1)∈F\documentclass[12pt]{minimal}
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\begin{document}$$H_{i}/{{\,\mathrm{Core}\,}}_{H_{i}}(H_{i-1}) \in \mathcal {F}$$\end{document} for every 1≤i≤n\documentclass[12pt]{minimal}
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\begin{document}$$1 \le i \le n$$\end{document}. If F=N\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}= \mathcal {N}$$\end{document}, the formation of all nilpotent groups, the K-N\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {N}$$\end{document}-subnormal subgroups of a group G are exactly the subnormal subgroups of G. The aim of this paper is to prove the following theorem: if F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document} is a subgroup-closed saturated lattice formation, then a subgroup H of a group G is K-F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document}-subnormal in G if and only if H is K-F\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {F}$$\end{document}-subnormal in <H,x>\documentclass[12pt]{minimal}
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\begin{document}$$<H,x>$$\end{document} for all x∈G\documentclass[12pt]{minimal}
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\begin{document}$$x \in G$$\end{document}. Some earlier results are consequence of this theorem.
