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 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]
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 .
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
- Direct factor
- Permutably complemented normal subgroup
- Strongly permutably complemented subgroup
- Conjugation-invariantly permutably complemented subgroup
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).
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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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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
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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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
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
This subgroup property does not satisfy 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
This subgroup property is quotient-transitive: the corresponding quotient property is transitive.
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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
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 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