We shall extend the research on power structure of finite p-groups in Mann (J Algebra 42:121–135, 1976) to locally nilpotentp-groups. Firstly, we obtain that a locally nilpotent Pi\documentclass[12pt]{minimal}
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\begin{document}$$P_i$$\end{document}-group G with |G:℧1(G)|<∞\documentclass[12pt]{minimal}
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\begin{document}$$|G:\mho _1(G)|< \infty $$\end{document} is an extension of a divisible abelian group by a finite p-group. Next we get the structure of infinite locally nilpotent p-groups which are not Pi\documentclass[12pt]{minimal}
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\begin{document}$$P_i$$\end{document}-groups, but all of whose proper infinite subgroups are Pi\documentclass[12pt]{minimal}
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\begin{document}$$P_i$$\end{document}-groups. Finally, we show that locally nilpotent Pi\documentclass[12pt]{minimal}
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\begin{document}$$P_i$$\end{document}-groups with all subgroups subnormal are nilpotent.
