Abstract
Wielandt’s criterion for the subnormality of a subgroup of a finite group is developed. For a set \(\pi=\{p_{1},p_{2},\mathinner{\ldotp\ldotp\ldotp},p_{n}\}\) and a partition \(\sigma=\{\{p_{1}\},\{p_{2}\},\mathinner{\ldotp\ldotp\ldotp},\{p_{n}\},\{\pi\}^ {\prime}\}\), it is proved that a subgroup \(H\) is \(\sigma\)-subnormal in a finite group \(G\) if and only if it is \(\{\{p_{i}\},\{p_{i}\}^{\prime}\}\)-subnormal in \(G\) for every \(i=1,2,\mathinner{\ldotp\ldotp\ldotp},n\). In particular, \(H\) is subnormal in \(G\) if and only if it is \(\{\{p\},\{p\}^{\prime}\}\)-subnormal in \(\langle H,H^{x}\rangle\) for every prime \(p\) and any element \(x\in G\).
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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 26, No. 3, pp. 211 - 218, 2020 https://doi.org/10.21538/0134-4889-2020-26-3-211-218.
Translated by E. Vasil’eva
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Sun, F., Yi, X. & Kamornikov, S.F. Criterion of Subnormality in a Finite Group: Reduction to Elementary Binary Partitions. Proc. Steklov Inst. Math. 313 (Suppl 1), S194–S200 (2021). https://doi.org/10.1134/S0081543821030202
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DOI: https://doi.org/10.1134/S0081543821030202