Base of a wreath product

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Definition

A subgroup $G$ of a group $K$ is termed a base of a wreath product if $K$ is expressible as an internal wreath product of $G$ by some subgroup $H$. In other words, there exists a subgroup $L$ of $K$ with $G$ that is a direct power of $G$ (with $G$ as one of the factors) and further, $K$ is the semidirect product of $L$ with a subgroup $H$ that acts on $L$ by permutations of the direct factors.

Relation with other properties

Stronger properties

• Direct factor: For proof of the implication, refer Direct factor implies base of a wreath product and for proof of its strictness (i.e. the reverse implication being false) refer Wreath product not implies direct factor.

Weaker properties

• AEP-subgroup
• Base of a wreath product with diagonal action
• Subset-conjugacy-closed subgroup: For full proof, refer: Base of a wreath product implies subset-conjugacy-closed
• Conjugacy-closed subgroup: For full proof, refer: Base of a wreath product implies conjugacy-closed
• 2-subnormal subgroup: For full proof, refer: Base of a wreath product implies 2-subnormal
• Right-transitively 2-subnormal subgroup: For proof of the implication, refer Base of a wreath product implies right-transitively 2-subnormal and for proof of its strictness (i.e. the reverse implication being false) refer Right-transitively 2-subnormal not implies base of a wreath product.
• Conjugate-permutable subgroup
• Right-transitively conjugate-permutable subgroup: For proof of the implication, refer Base of a wreath product implies right-transitively conjugate-permutable and for proof of its strictness (i.e. the reverse implication being false) refer Right-transitively conjugate-permutable not implies base of a wreath product.
• TI-subgroup: For full proof, refer: Base of a wreath product implies TI

Other related properties

• Base diagonal of a wreath product

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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If $H$ is the base of a wreath product in a group $K$, and $K$ is the base of a wreath product in a group $G$, then $H$ is the base of a wreath product in $G$. For full proof, refer: Base of a wreath product is transitive

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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Every group is the base of a wreath product in itself -- the wreath product with the trivial group. In contrast, the trivial subgroup is the base of a wreath product -- the wreath product with the whole group acting trivially on it.

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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If $H$ is the base of a wreath product in $G$, then $H$ is also the base of a wreath product in any intermediate subgroup. For full proof, refer: Base of a wreath product satisfies intermediate subgroup condition