A subgroup \documentclass[12pt]{minimal}
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\begin{document}$$H$$\end{document} of a group \documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document} is called \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{P }$$\end{document}-subnormal in \documentclass[12pt]{minimal}
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\begin{document}$$G$$\end{document} whenever either \documentclass[12pt]{minimal}
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\begin{document}$$H=G$$\end{document} or there is a chain of subgroups \documentclass[12pt]{minimal}
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\begin{document}$$H=H_0\subset H_1\subset \cdots \subset H_n=G$$\end{document} such that \documentclass[12pt]{minimal}
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\begin{document}$$|H_i:H_{i-1}|$$\end{document} is a prime for all \documentclass[12pt]{minimal}
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\begin{document}$$i$$\end{document}. In this paper we study groups with \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{P }$$\end{document}-subnormal 2-maximal subgroups, and groups with \documentclass[12pt]{minimal}
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\begin{document}$$\mathbb{P }$$\end{document}-subnormal primary cyclic subgroups.