Base of a wreath product

Definition
A subgroup $$G$$ of a group $$K$$ is termed a base of a wreath product if $$K$$ is expressible as an defining ingredient::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.

Stronger properties

 * Weaker than::Direct factor:

Weaker properties

 * Stronger than::AEP-subgroup
 * Stronger than::Base of a wreath product with diagonal action
 * Stronger than::Subset-conjugacy-closed subgroup:
 * Stronger than::Conjugacy-closed subgroup:
 * Stronger than::2-subnormal subgroup:
 * Stronger than::Right-transitively 2-subnormal subgroup:
 * Stronger than::Conjugate-permutable subgroup
 * Stronger than::Right-transitively conjugate-permutable subgroup:
 * Stronger than::TI-subgroup:

Other related properties

 * Base diagonal of a wreath product

Metaproperties
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 $$G$$.

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.

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.