Permuting transfer-closed conjugacy-closed-to-retract

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 defining ingredient::permuting subgroups: $$H_iK_i = K_iH_i$$.

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

Stronger properties

 * Weaker than::Sylow subgroup:

Weaker properties

 * Stronger than::Intermediately conjugacy-closed-to-retract
 * Stronger than::Permuting transfer-closed central factor-to-direct factor
 * Stronger than::Intermediately central factor-to-direct factor