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 $$\pi=\{p_{1},p_{2},\mathinner{\ldotp\ldotp\ldotp},p_{n}\}$$ 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 $$x\in G$$ .