Wielandt's criterion for the subnormality of a subgroup in a finite group is developed. For a set $\pi =\{p_1,p_2,\ldots,p_n\}$ and a partition $\sigma=\{\{p_1\},\{p_2\},\ldots,\{p_n\},\{\pi\}^{'}\}$, it is proved that a subgroup $H$ is $\sigma$-subnormal in a finite group $G$ if and only if it is $\{\{p_i\},\{p_i\}^{'}\}$-subnormal in $G$ for every $i=1,2,\ldots,n$. In particular, $H$ is subnormal in $G$ if and only if it is $\{\{p\},\{p\}^{'}\}$-subnormal in $\langle H,H^x\rangle$ for every prime $p$ and any element $x\in G$.
Keywords: finite group, subnormal subgroup, $\sigma$-subnormal subgroup, elementary binary partition
Received June 4, 2020
Revised June 30, 2020
Accepted July 3, 2020
Fenfen Sun, Zhejiang Sci-Tech University, Hangzhou, P. R. China, e-mail: sun4624@163.com
Xiaolan Yi, Zhejiang Sci-Tech University, Hangzhou, P. R. China, e-mail: yixiaolan2005@126.com
Sergei Fedorovich Kamornikov, Dr. Phys.-Math. Sci., Prof., Francisk Skorina Gomel State University, 246019, Gomel, Republic of Belarus. e-mail: sfkamornikov@mail.ru
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Cite this article as: F. Sun, X. Yi, S.F. Kamornikov. Criterion of subnormality in a finite group: Reduction to elementary binary partitions, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2020, vol. 26, no. 3, pp. 211–218; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2021, Vol. 313, Suppl. 1, pp. S194–S200.
