# Permutably complemented subgroup

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 definitionVIEW: Definitions built on this | Facts about this: (factscloselyrelated to Permutably complemented subgroup, all facts related to Permutably complemented subgroup) |Survey articles about this | Survey articles about definitions built on this

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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 of a group is said to be **permutably complemented** if there is a subgroup of such that and is trivial.

is termed a permutable complement to .

## 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

View other monadic second-order subgroup properties

## Relation with other properties

### Stronger properties

- Retract
- Direct factor
- Permutably complemented normal subgroup
- Strongly permutably complemented subgroup
- Conjugation-invariantly permutably complemented subgroup

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

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT 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 subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

If is a permutably complemented subgroup of and is a permutably complemented subgroup of , need not be permutably complemented in . The reason is that the permutable complement of in need *not* permute with the permutable complement of in . `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 doesnotsatisfy the transfer condition

If is a permutably complemented subgroup of , and is some subgroup of , need not be permutably complemented in . `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 are such that is a permutably complemented normal subgroup of and is a permutably complemented subgroup of . Then, is also permutably complemented in . `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

View other subgroup properties satisfying image condition

If is a surjective homomorphism, and is a permutably complemented subgroup of , is a permutably complemented subgroup of . `For full proof, refer: Permutably complemented satisfies image condition`