Criterion of Subnormality in a Finite Group: Reduction to Elementary Binary Partitions
2021
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 $$x\in G$$
.
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