Base of a wreath product implies right-transitively conjugate-permutable

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., base of a wreath product) must also satisfy the second subgroup property (i.e., right-transitively conjugate-permutable subgroup)
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This article describes a computation relating the result of the Composition operator (?) on two known subgroup properties (i.e., Conjugate-permutable subgroup (?) and Base of a wreath product (?)), to another known subgroup property (i.e., Right-transitively conjugate-permutable subgroup (?))
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Statement

Property-theoretic statement

The following equivalent statements are true:

1. The subgroup property of being the base of a wreath product implies, i.e., is stronger than, the subgroup property of being a right-transitively conjugate-permutable subgroup.
2. Applying the composition operator to the property of being a conjugate-permutable subgroup and the property of being the base of a wreath product, yields a property stronger than the property of being a conjugate-permutable subgroup.

Conjugate-permutable $*$ Base of a wreath product $\leq$ Conjugate-permutable

Verbal statement

The following equivalent statements are true:

1. Any base of a wreath product in a group is a right-transitively conjugate-permutable subgroup.
2. A conjugate-permutable subgroup of the base of a wreath product is conjugate-permutable in the whole group.

Statement with symbols

The following equivalent statements are true:

1. If $K$ is the base of a wreath product in a group $G$, then $K$ is a right-transitively conjugate-permutable subgroup of $G$.
2. If $H$ is a conjugate-permutable subgroup of $K$ and $K$ is the base of a wreath product in $G$, then $H$ is conjugate-permutable in $G$.

Related facts

• Base of a wreath product implies right-transitively 2-subnormal
• Base of a wreath product is transitive

Proof

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