Recall that a subgroup A of a group G is called G-permutable in G if for every subgroup B of G there exists an element x∈G\documentclass[12pt]{minimal}
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\begin{document}$$x\in G$$\end{document} such that A and Bx\documentclass[12pt]{minimal}
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\begin{document}$$B^x$$\end{document} commute. The following question was posed in the Kourovka Notebook: is there an integer n such that for all m>n\documentclass[12pt]{minimal}
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\begin{document}$$m>n$$\end{document} the alternating group Am\documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{A}\,}}_m$$\end{document} has no non-trivial Am\documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{A}\,}}_m$$\end{document}-permutable subgroups? We give a positive answer to this question. Moreover, in the case of prime p we prove that Ap\documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{A}\,}}_p$$\end{document} has no non-trivial Ap\documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{A}\,}}_p$$\end{document}-permutable subgroups except p=5\documentclass[12pt]{minimal}
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\begin{document}$$p=5$$\end{document}.
