Permuting transfer-closed normal-to-complemented subgroup

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Definition

A subgroup $H$ of a group $G$ is termed permuting transfer-closed normal-to-complemented in $G$ if the following is true.

Suppose $K_{1},K_{2},\dots K_{n}$ and $H=H_{1},H_{2},\dots H_{n},H_{n+1}$ are subgroups such that:

$H_{i+1}=H_{i}\cap K_{i}$ .
$H_{i}$ and $K_{i}$ are permuting subgroups: $H_{i}K_{i}=K_{i}H_{i}$ .

Then, if $H_{n+1}$ is a normal subgroup of $K_{n}$ , then $H_{n+1}$ is a complemented normal subgroup of $K_{n}$ .

Relation with other properties

Stronger properties

Hall subgroup: For full proof, refer: Hall implies permuting transfer-closed normal-to-complemented

Weaker properties