Groupprops, The Group Properties Wiki (pre-alpha)

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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.
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VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions |
VIEW FACTS THAT: use, or start with, a subgroup satisfying the property| prove, or show that, a subgroup satisfies the property
RANDOM SUBGROUP PROPERTY: Fully characteristic subgroup: A subgroup such that any endomorphism of the whole group takes the subgroup to within itself.

Contents 1 Definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties 2.3 Other related properties 3 Metaproperties 3.1 Transitivity 3.2 Trimness 3.3 Intermediate subgroup condition

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

• 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

Other related properties

• Base diagonal of a wreath product

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: |
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: |
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