Permuting transfer-closed conjugacy-closed-to-retract

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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 conjugacy-closed subgroup of $K_{n}$ , then $H_{n + 1}$ is a retract of $K_{n}$ : it has a normal complement in $K_{n}$ .

Relation with other properties

Stronger properties

Sylow subgroup: For full proof, refer: Sylow implies permuting transfer-closed conjugacy-closed-to-retract

Weaker properties