Base of a wreath product

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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 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

Weaker properties

Other related properties

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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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 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.
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ABOUT 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