# 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]

## Contents

## Definition

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

## 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 THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT 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 is the base of a wreath product in a group , and is the base of a wreath product in a group , then is the base of a wreath product in . `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.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If is the base of a wreath product in , then 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`