Quotient-powering-invariant subgroup: Difference between revisions
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{{subgroup property}} | {{subgroup property}} | ||
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{{subgroup property conjunction|powering-invariant subgroup|normal subgroup satisfying the subgroup-to-quotient powering-invariance implication}} | |||
==Definition== | ==Definition== | ||
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| [[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::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>. | ||
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| [[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>. | |||
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| [[Weaker than::finite normal subgroup]] || the normal subgroup is finite || [[finite normal implies quotient-powering-invariant]] || || | | [[Weaker than::finite normal subgroup]] || the normal subgroup is finite || [[finite normal implies quotient-powering-invariant]] || || | ||
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| [[Weaker than:: | | [[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}} | ||
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| [[Weaker than::complemented normal subgroup]] || normal subgroup with a (possibly non-normal) complement || [[complemented normal implies quotient-powering-invariant]] || || {{intermediate notions short|quotient-powering-invariant subgroup|complemented normal subgroup}} | | [[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}} | ||
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| [[Weaker than::direct factor]] || normal subgroup with normal complement || (via complemented normal) || (via complemented normal) || {{intermediate notions short|quotient-powering-invariant subgroup|direct factor}} | |||
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| [[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 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}} | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[Stronger than::powering-invariant subgroup]] || || [[quotient-powering-invariant implies powering-invariant]] || [[powering-invariant not implies quotient-powering-invariant]] || {{intermediate notions short|powering-invariant subgroup|quotient-powering-invariant subgroup}} | | [[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}} | ||
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| [[Stronger than::normal 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}} | ||
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| [[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}} | |||
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| [[Stronger than::normal subgroup]] || || (by definition) || (via powering-invariant normal) || {{intermediate notions short|normal subgroup|quotient-powering-invariant subgroup}} | |||
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===Properties whose conjunction with powering-invariance implies quotient-powering-invariance=== | ===Properties whose conjunction with powering-invariance implies quotient-powering-invariance=== | ||
The relevant subgroup property is [[normal subgroup | 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]]. | |||
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]
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 of a group is termed a quotient-powering-invariant subgroup if, for any prime number such that is a powered for , the quotient group is also powered for .
Metaproperties
| Metaproperty name | Satisfied? | Proof | Statement with symbols |
|---|---|---|---|
| quotient-transitive subgroup property | Yes | quotient-powering-invariance is quotient-transitive | If are such that is quotient-powering-invariant in and is quotient-powering-invariant in , then is quotient-powering-invariant in . |
| union-closed subgroup property | Yes | quotient-powering-invariance is union-closed | If are all quotient-powering-invariant subgroups of a group , and their set-theoretic union is a subgroup , then is also a quotient-powering-invariant subgroup of . |
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 | Meaning | 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.