# Base of a wreath product

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## 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.

## 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.
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

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).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

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.
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