Permutably complemented subgroup: Difference between revisions

Latest revision as of 18:49, 20 May 2009

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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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
View other monadic second-order subgroup properties

Relation with other properties

Stronger properties

Weaker properties

Lattice-complemented subgroup

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 condition
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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
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 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 $H\leq K\leq G$ 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
View other subgroup properties satisfying image condition

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

@@ Line 1: / Line 1: @@
+{{semibasicdef}}
 {{subgroup property}}
-{{semibasicdef}}
 ==Definition==
@@ Line 15: / Line 14: @@
 A subgroup <math>H</math> of a group <math>G</math> is said to be '''permutably complemented''' if there is a subgroup <math>K</math> of <math>G</math> such that <math>HK = KH = G</math> and <math>H \cap K</math> is trivial.
-<math>K</math> is termed a [[permutable complement]] to <math>H</math>.
+<math>K</math> is termed a [[permutable complements|permutable complement]] to <math>H</math>.
+==Formalisms==
+{{wikilocal-section}}
+{{monadic second-order subgroup property}}
 ==Relation with other properties==
@@ Line 21: / Line 27: @@
 ===Stronger properties===
-* [[Retract]]
+* [[Weaker than::Retract]]
-* [[Direct factor]]
+* [[Weaker than::Direct factor]]
-* [[Permutably complemented normal subgroup]]
+* [[Weaker than::Permutably complemented normal subgroup]]
-* [[Strongly permutably complemented subgroup]]
+* [[Weaker than::Strongly permutably complemented subgroup]]
+* [[Weaker than::Conjugation-invariantly permutably complemented subgroup]]
 ===Weaker properties===
-* [[Lattice-complemented subgroup]]
+* [[Stronger than::Lattice-complemented subgroup]]
 ==Metaproperties==
@@ Line 36: / Line 43: @@
 The trivial subgroup and the whole group are permutable complements of each other, hence both are permutably complemented subgroups.
-===Transitivity===
+{{intsubcondn}}
+if <math>H</math> is a permutably complemented subgroup of <math>G</math>, then <math>H</math> is also permutably complemented in every intermediate subgroup <math>K</math> of <math>G</math>. This follows from the [[modular property of groups]]. {{further|[[Permutably complemented satisfies intermediate subgroup condition]]}}
+{{intransitive}}
+If <math>H</math> is a permutably complemented subgroup of <math>K</math> and <math>K</math> is a permutably complemented subgroup of <math>G</math>, <math>H</math> need not be permutably complemented in <math>G</math>. The reason is that the permutable complement of <math>H</math> in <math>K</math> need ''not'' permute with the permutable complement of <math>K</math> in <math>G</math>. {{proofat|[[Permutably complemented is not transitive]]}}
+{{not intersection-closed}}
+An intersection of permutably complemented subgroups need not be permutably complemented. {{proofat|[[Permutably complemented is not intersection-closed]]}}
+{{not transfercondn}}
+If <math>H</math> is a permutably complemented subgroup of <math>G</math>, and <math>K</math> is some subgroup of <math>G</math>, <math>H \cap K</math> need not be permutably complemented in <math>K</math>. {{proofat|[[Permutably complemented does not satisfy transfer condition]]}}
+{{quot-transitive}}
-It does not seem necessary that a permutably complemented subgroup of a permutably complemented subgroup should be permutably complemented. However, it is true that any permutably complemented subgroup of a [[strongly permutably complemented subgroup]] is permutably complemented.
+Suppose <math>H \le K \le G</math> are such that <math>H</math> is a permutably complemented normal subgroup of <math>G</math> and <math>K/H</math> is a permutably complemented subgroup of <math>G/H</math>. Then, <math>K</math> is also permutably complemented in <math>G</math>. {{further|[[Permutably complemented is quotient-transitive]]}}
-===Intersection-closedness===
+{{imagecondn}}
-An intersection of permutably complemented subgroups need not be permutably complemented.
+If <math>\varphi:G \to K</math> is a surjective homomorphism, and <math>H</math> is a permutably complemented subgroup of <math>G</math>, <math>\varphi(H)</math> is a permutably complemented subgroup of <math>K</math>. {{proofat|[[Permutably complemented satisfies image condition]]}}