Criterion of Subnormality in a Finite Group: Reduction to Elementary Binary Partitions

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 $x\in G$.