Base of a wreath product

From Groupprops
Jump to: navigation, search
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.

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.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
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.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

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