Quotient-powering-invariant subgroup: Difference between revisions

Latest revision as of 01:38, 1 April 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]
|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Quotient-powering-invariant subgroup, all facts related to Quotient-powering-invariant subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: powering-invariant subgroup and normal subgroup satisfying the subgroup-to-quotient powering-invariance implication
View other subgroup property conjunctions | view all subgroup properties

Definition

A normal subgroup $H$ of a group $G$ is termed a quotient-powering-invariant subgroup if, for any prime number $p$ such that $G$ is a powered for $p$ , the quotient group $G / H$ is also powered for $p$ .

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
quotient-transitive subgroup property	Yes	quotient-powering-invariance is quotient-transitive	If $H \leq K \leq G$ are such that $H$ is quotient-powering-invariant in $G$ and $K / H$ is quotient-powering-invariant in $G / H$ , then $K$ is quotient-powering-invariant in $G$ .
union-closed subgroup property	Yes	quotient-powering-invariance is union-closed	If $H_{i}, i \in I$ are all quotient-powering-invariant subgroups of a group $G$ , and their set-theoretic union $⋃_{i \in I} H_{i}$ is a subgroup $H$ , then $H$ is also a quotient-powering-invariant subgroup of $G$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup of finite group	the whole group is finite
normal subgroup of periodic group	every element in the whole group has finite order
normal subgroup of finite index	the quotient group is finite	normal of finite index implies quotient-powering-invariant
finite normal subgroup	the normal subgroup is finite	finite normal implies quotient-powering-invariant
endomorphism kernel	normal subgroup that is the kernel of an endomorphism	endomorphism kernel implies quotient-powering-invariant	any normal subgroup of a finite group that is not an endomorphism kernel works.	\|FULL LIST, MORE INFO
complemented normal subgroup	normal subgroup with a (possibly non-normal) complement	(via endomorphism kernel, see also direct proof)		\|FULL LIST, MORE INFO
direct factor	normal subgroup with normal complement	(via complemented normal)	(via complemented normal)	\|FULL LIST, MORE INFO
characteristic subgroup of abelian group	characteristic subgroup and the whole group is an abelian group	characteristic subgroup of abelian group is quotient-powering-invariant		\|FULL LIST, MORE INFO

Weaker properties

Property	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
powering-invariant normal subgroup	quotient-powering-invariant implies powering-invariant	powering-invariant and normal not implies quotient-powering-invariant	\|FULL LIST, MORE INFO
powering-invariant subgroup	quotient-powering-invariant implies powering-invariant	(via powering-invariant normal)	\|FULL LIST, MORE INFO
normal subgroup satisfying the subgroup-to-quotient powering-invariance implication			\|FULL LIST, MORE INFO
normal subgroup	(by definition)	(via powering-invariant normal)	\|FULL LIST, MORE INFO

Properties whose conjunction with powering-invariance implies quotient-powering-invariance

The relevant subgroup property is normal subgroup satisfying the subgroup-to-quotient powering-invariance implication. In fact, the conjunction of this with powering-invariant subgroup precisely gives quotient-powering-invariant subgroup.

This property is implied both by being a central subgroup and by being a normal subgroup contained in the hypercenter.

@@ Line 1: / Line 1: @@
 {{subgroup property}}
+{{wikilocal}}
+{{subgroup property conjunction|powering-invariant subgroup|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication}}
 ==Definition==
 A [[normal subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed a '''quotient-powering-invariant subgroup''' if, for any [[prime number]] <math>p</math> such that <math>G</math> is a [[powered group for a set of primes|powered for]] <math>p</math>, the [[quotient group]] <math>G/H</matH> is also powered for <math>p</math>.
+==Metaproperties==
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::quotient-transitive subgroup property]] || Yes || [[quotient-powering-invariance is quotient-transitive]] || If <math>H \le K \le G</math> are such that <math>H</math> is quotient-powering-invariant in <math>G</math> and <math>K/H</matH> is quotient-powering-invariant in <math>G/H</math>, then <math>K</math> is quotient-powering-invariant in <math>G</math>.
+|-
+| [[satisfies metaproperty::union-closed subgroup property]] || Yes || [[quotient-powering-invariance is union-closed]] || If <math>H_i, i \in I</math> are all quotient-powering-invariant subgroups of a group <math>G</math>, and their set-theoretic union <math>\bigcup_{i \in I} H_i</math> is a subgroup <math>H</math>, then <math>H</math> is also a quotient-powering-invariant subgroup of <math>G</math>.
+|}
 ==Relation with other properties==
@@ Line 12: / Line 23: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Weaker than::normal subgroup of finite index]] || || || ||
+| [[Weaker than::normal subgroup of finite group]] || the whole group is finite || || ||
+|-
+| [[Weaker than::normal subgroup of periodic group]] || every element in the whole group has finite order || || ||
+|-
+| [[Weaker than::normal subgroup of finite index]] || the quotient group is finite || [[normal of finite index implies quotient-powering-invariant]] || ||
+|-
+| [[Weaker than::finite normal subgroup]] || the normal subgroup is finite || [[finite normal implies quotient-powering-invariant]] || ||
+|-
+| [[Weaker than::endomorphism kernel]] || normal subgroup that is the kernel of an [[endomorphism]] || [[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}}
+|-
+| [[Weaker than::complemented normal subgroup]] || normal subgroup with a (possibly non-normal) complement || (via endomorphism kernel, see also [[complemented normal implies quotient-powering-invariant|direct proof]]) || || {{intermediate notions short|quotient-powering-invariant subgroup|complemented normal subgroup}}
+|-
+| [[Weaker than::direct factor]] || normal subgroup with normal complement || (via complemented normal) || (via complemented normal) || {{intermediate notions short|quotient-powering-invariant subgroup|direct factor}}
+|-
+| [[Weaker than::characteristic subgroup of abelian group]] || [[characteristic subgroup]] and the whole group is an [[abelian group]] || [[characteristic subgroup of abelian group is quotient-powering-invariant]] || || {{intermediate notions short|quotient-powering-invariant subgroup|characteristic subgroup of abelian group}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::powering-invariant normal subgroup]] || || [[quotient-powering-invariant implies powering-invariant]] || [[powering-invariant and normal not implies quotient-powering-invariant]] || {{intermediate notions short|powering-invariant normal subgroup|quotient-powering-invariant subgroup}}
+|-
+| [[Stronger than::powering-invariant subgroup]] || || [[quotient-powering-invariant implies powering-invariant]] || (via powering-invariant normal) || {{intermediate notions short|powering-invariant subgroup|quotient-powering-invariant subgroup}}
+|-
+| [[Stronger than::normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]] || || || || {{intermediate notions short|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication|quotient-powering-invariant subgroup}}
+|-
+| [[Stronger than::normal subgroup]] || || (by definition) || (via powering-invariant normal) || {{intermediate notions short|normal subgroup|quotient-powering-invariant subgroup}}
 |}
+===Properties whose conjunction with powering-invariance implies quotient-powering-invariance===
+The relevant subgroup property is [[normal subgroup satisfying the subgroup-to-quotient powering-invariance implication]]. In fact, the conjunction of this with [[powering-invariant subgroup]] precisely gives [[quotient-powering-invariant subgroup]].
+This property is implied both by being a [[central subgroup]] and by being a [[normal subgroup contained in the hypercenter]].