We reduce
the solution of the Kegel–Wielandt \documentclass[12pt]{minimal}
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\begin{document}$ \sigma $\end{document}-problem
for an arbitrary partition \documentclass[12pt]{minimal}
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\begin{document}$ \sigma $\end{document} of the set of all primes
to its solution in the class of all \documentclass[12pt]{minimal}
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\begin{document}$ \sigma $\end{document}-complete
simple nonabelian groups.
We also give a sufficient criterion for the
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\begin{document}$ \sigma $\end{document}-subnormality of a subgroup in a finite group for a partition \documentclass[12pt]{minimal}
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\begin{document}$ \sigma $\end{document} in which 2 and 3 belong to the same component.