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

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 $x\in G$.