Permutably complemented subgroup

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This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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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

Symbol-free definition

A subgroup of a group is said to be permutably complemented if there is another subgroup that intersects it trivially and such that the product group of these groups is the whole group.

This other subgroup is termed a permutable complement.

Definition with symbols

A subgroup H of a group G is said to be permutably complemented if there is a subgroup K of G such that HK = KH = G and H \cap K is trivial.

K is termed a permutable complement to H.

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

Monadic second-order description

This subgroup property is a monadic second-order subgroup property, viz., it has a monadic second-order description in the theory of groups
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Relation with other properties

Stronger properties

Weaker properties

Metaproperties

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

The trivial subgroup and the whole group are permutable complements of each other, hence both are permutably complemented subgroups.

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 H is a permutably complemented subgroup of G, then H is also permutably complemented in every intermediate subgroup K of G. This follows from the modular property of groups. Further information: Permutably complemented satisfies intermediate subgroup condition

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
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ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

If H is a permutably complemented subgroup of K and K is a permutably complemented subgroup of G, H need not be permutably complemented in G. The reason is that the permutable complement of H in K need not permute with the permutable complement of K in G. For full proof, refer: Permutably complemented is not transitive

Intersection-closedness

This subgroup property is not intersection-closed, viz., it is not true that an intersection of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not intersection-closed

An intersection of permutably complemented subgroups need not be permutably complemented. For full proof, refer: Permutably complemented is not intersection-closed

Transfer condition

This subgroup property does not satisfy the transfer condition

If H is a permutably complemented subgroup of G, and K is some subgroup of G, H \cap K need not be permutably complemented in K. For full proof, refer: Permutably complemented does not satisfy transfer condition

Quotient-transitivity

This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
View a complete list of quotient-transitive subgroup properties

Suppose H \le K \le G are such that H is a permutably complemented normal subgroup of G and K/H is a permutably complemented subgroup of G/H. Then, K is also permutably complemented in G. Further information: Permutably complemented is quotient-transitive

Image condition

YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
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If \varphi:G \to K is a surjective homomorphism, and H is a permutably complemented subgroup of G, \varphi(H) is a permutably complemented subgroup of K. For full proof, refer: Permutably complemented satisfies image condition