Geodesic
Criterion of Subnormality in a Finite Group: Reduction to Elementary Binary Partitions
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 3, pp. 211-218 Cet article a éte moissonné depuis la source Math-Net.Ru

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Wielandt's criterion for the subnormality of a subgroup of 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. 
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F. Sun; X. Yi; S. F. Kamornikov. Criterion of Subnormality in a Finite Group: Reduction to Elementary Binary Partitions. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 3, pp. 211-218. http://geodesic.mathdoc.fr/item/TIMM_2020_26_3_a17/

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