Endomorphism kernel: Difference between revisions

Latest revision as of 20:21, 16 February 2013

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]
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Definition

Equivalent definitions in tabular format

No.	Shorthand	A subgroup of a group is termed an endomorphism kernel if ...	A subgroup $H$ of a group $G$ is termed an endomorphism kernel in $G$ if ...
1	normal, quotient isomorphic to subgroup	it is normal in the whole group and its quotient group is isomorphic to some subgroup of the whole group.	$H$ is a normal subgroup of $G$ and there is a subgroup $M$ of $G$ such that the quotient group $G / H$ is isomorphic to $M$ .
2	endomorphism kernel	there is an endomorphism of the whole group whose kernel is precisely the subgroup.	there is an endomorphism $σ$ of $G$ such that the kernel of $σ$ is $H$

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	No	endomorphism kernel is not transitive	It is possible to have group $H \leq K \leq G$ such that $H$ is an endomorphism kernel in $K$ and $K$ is an endomorphism kernel in $G$ but $H$ is not an endomorphism kernel in $G$ .
intermediate subgroup condition	No	endomorphism kernel does not satisfy intermediate subgroup condition	It is possible to have groups $H \leq K \leq G$ such that $H$ is an endomorphism kernel in $G$ but is not an endomorphism kernel in $K$ .
quotient-transitive subgroup property	Yes	endomorphism kernel is quotient-transitive	Suppose $H \leq K \leq G$ are groups such that $H$ is an endomorphism kernel in $G$ and $K / H$ is an endomorphism kernel in $G / H$ . Then, $K$ is an endomorphism kernel in $G$ .
trim subgroup property	Yes	obvious	in any group $G$ , the trivial subgroup and the whole group $G$ are endomorphism kernels.

Relation with other properties

Stronger properties

Property	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
complemented normal subgroup	complemented normal implies endomorphism kernel	endomorphism kernel not implies complemented normal	\|FULL LIST, MORE INFO
direct factor			\|FULL LIST, MORE INFO
subgroup of finite abelian group	follows from subgroup lattice and quotient lattice of finite abelian group are isomorphic	(trivial subgroup, whole group are endomorphism kernels even in non-abelian groups)	\|FULL LIST, MORE INFO

Weaker properties

Property	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
quotient-powering-invariant subgroup	endomorphism kernel implies quotient-powering-invariant	any normal subgroup of a finite group that is not an endomorphism kernel works.	\|FULL LIST, MORE INFO
powering-invariant normal subgroup	(via quotient-powering-invariant)	(via quotient-powering-invariant)	\|FULL LIST, MORE INFO
powering-invariant subgroup	(via quotient-powering-invariant)	(via quotient-powering-invariant)	\|FULL LIST, MORE INFO
normal subgroup	(by definition)	normal not implies endomorphism kernel	\|FULL LIST, MORE INFO

Effect of property operators

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)

Operator	Meaning	Result of application	Proof
potentially operator	endomorphism kernel in some larger group	normal subgroup	normal implies potential endomorphism kernel
intermediately operator	endomorphism kernel in every intermediate subgroup	intermediately endomorphism kernel	(by definition)

@@ Line 5: / Line 5: @@
 ==Definition==
-===Symbol-free definition===
+===Equivalent definitions in tabular format===
-A [[subgroup]] of a [[group]] is termed an '''endomorphic kernel''' if it satisfies the following equivalent conditions:
+{| class="sortable" border="1"
+! No. !! Shorthand !! A [[subgroup]] of a [[group]] is termed an endomorphism kernel if ... !! A subgroup <math>H</math> of a [[group]] <math>G</math> is termed an endomorphism kernel in <math>G</math> if ...
+|-
+| 1 || normal, quotient isomorphic to subgroup || it is [[normal subgroup|normal]] in the whole group and its [[quotient group]] is isomorphic to some subgroup of the whole group.  || <math>H</math> is a normal subgroup of <math>G</math> and there is a subgroup <math>M</math> of <math>G</math> such that the [[quotient group]] <math>G/H</math> is isomorphic to <math>M</math>.
+|-
+| 2 || endomorphism kernel || there is an endomorphism of the whole group whose kernel is precisely the subgroup. || there is an endomorphism <math>\sigma</math> of <math>G</math> such that the kernel of <math>\sigma</math> is <math>H</math>
+|}
-* It is [[normal subgroup|normal]] and there is a subgroup of the group isomorphic to the quotient group
+==Metaproperties==
-* There is an endomorphism of the group whose kernel is the given subgroup
-===Definition with symbols===
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''endomorphic kernel''' if it satisfies the following conditions:
-* There is a subgroup <math>K</math> of <math>G</math> such that <math>G/H \cong K</math>
-* There is an endomorphism <math>\rho</math> of <math>G</math> such that the kernel of <math>\rho</math> is <math>H</math>
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[dissatisfies metaproperty::transitive subgroup property]] || No || [[endomorphism kernel is not transitive]] || It is possible to have group <math>H \le K \le G</math> such that <math>H</math> is an endomorphism kernel in <math>K</math> and <math>K</math> is an endomorphism kernel in <math>G</math> but <math>H</math> is not an endomorphism kernel in <math>G</math>.
+|-
+| [[dissatisfies metaproperty::intermediate subgroup condition]] || No || [[endomorphism kernel does not satisfy intermediate subgroup condition]] || It is possible to have groups <math>H \le K \le G</math> such that <math>H</math> is an endomorphism kernel in <math>G</math> but is not an endomorphism kernel in <math>K</math>.
+|-
+| [[satisfies metaproperty::quotient-transitive subgroup property]] || Yes || [[endomorphism kernel is quotient-transitive]] || Suppose <math>H \le K \le G</math> are [[group]]s such that <math>H</math> is an endomorphism kernel in <math>G</math> and <math>K/H</math> is an endomorphism kernel in <math>G/H</math>. Then, <math>K</math> is an endomorphism kernel in <math>G</math>.
+|-
+| [[satisfies metaproperty::trim subgroup property]] || Yes || obvious || in any group <math>G</math>, the trivial subgroup and the whole group <math>G</math> are endomorphism kernels.
+|}
 ==Relation with other properties==
 ===Stronger properties===
-* [[Complemented normal subgroup]]
+{| class="sortable" border="1"
-* [[Direct factor]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::complemented normal subgroup]] || || [[complemented normal implies endomorphism kernel]] || [[endomorphism kernel not implies complemented normal]] || {{intermediate notions short|endomorphism kernel|complemented normal subgroup}}
+|-
+| [[Weaker than::direct factor]] || || || || {{intermediate notions short|endomorphism kernel|direct factor}}
+|-
+| [[Weaker than::subgroup of finite abelian group]] || || follows from [[subgroup lattice and quotient lattice of finite abelian group are isomorphic]] || (trivial subgroup, whole group are endomorphism kernels even in non-abelian groups) || {{intermediate notions short|endomorphism kernel|subgroup of finite abelian group}}
+|}
 ===Weaker properties===
-* [[Normal subgroup]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-==Metaproperties==
+|-
+| [[Stronger than::quotient-powering-invariant subgroup]] || || [[endomorphism kernel implies quotient-powering-invariant]] || any normal subgroup of a finite group that is not an endomorphism kernel works. || {{intermediate notions short|quotient-powering-invariant subgroup|endomorphism kernel}}
-{{quot-transitive}}
+|-
+| [[Stronger than::powering-invariant normal subgroup]] || || ([[quotient-powering-invariant implies powering-invariant|via quotient-powering-invariant]]) || (via quotient-powering-invariant) || {{intermediate notions short|powering-invariant normal subgroup|endomorphism kernel}}
-If <math>N</math> is an endomorphic kernel in <math>G</math>, and <math>M</math> is a subgroup containing <math>N</math> such that <math>M/N</math> is an endomorphic kernel in <math>G/N</math>.
+|-
+| [[Stronger than::powering-invariant subgroup]] || || ([[quotient-powering-invariant implies powering-invariant|via quotient-powering-invariant]]) || (via quotient-powering-invariant) || {{intermediate notions short|powering-invariant subgroup|endomorphism kernel}}
+|-
+| [[Stronger than::normal subgroup]] || || (by definition) || [[normal not implies endomorphism kernel]] || {{intermediate notions short|normal subgroup|endomorphism kernel}}
+|}
-The proof of this follows by simply composing the two endomorphisms.
+==Effect of property operators==
-{{trim}}
+{{wikilocal-section}}
-The trivial subgroup is the kernel of the [[identity map]], while the whole group is the kernel of the trivial endomorphism.
+{| class="sortable" border="1"
+! Operator !! Meaning !! Result of application !! Proof
+|-
+| [[potentially operator]] || endomorphism kernel in some larger group || [[normal subgroup]] || [[normal implies potential endomorphism kernel]]
+|-
+| [[intermediately operator]] || endomorphism kernel in every intermediate subgroup || [[intermediately endomorphism kernel]] || (by definition)
+|}