Difference between revisions of "Finite subnormal subgroup"

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(New page: {{group-subgroup property conjunction|subnormal subgroup|finite group}} ==Definition== A subgroup of a group is termed a '''finite subnormal subgroup''' if it is [[finite group|f...)
 
 
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===Stronger properties===
 
===Stronger properties===
  
* [[Weaker than::Finite characteristic subgroup]]
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{| class="sortable" border="1"
* [[Weaker than::Finite normal subgroup]]
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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|-
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| [[Weaker than::Finite characteristic subgroup]] || the subgroup is both a [[finite group]] and a [[characteristic subgroup]] || (characteristic implies subnormal, via normal) || use [[normal not implies characteristic]] (finite examples) || {{intermediate notions short|finite subnormal subgroup|finite characteristic subgroup}}
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|-
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| [[Weaker than::Finite normal subgroup]] || the subgroup is both a [[finite group]] and a [[normal subgroup]] || (normal implies subnormal) || [[normality is not transitive]] (finite examples) || {{intermediate notions short|finite subnormal subgroup|finite normal subgroup}}
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|-
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| [[Weaker than::Subnormal subgroup of finite group]] || [[subnormal subgroup]] where the whole group is a [[finite group]] || || || {{intermediate notions short|finite subnormal subgroup|subnormal subgroup of finite group}}
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|}
  
 
===Weaker properties===
 
===Weaker properties===
  
* [[Stronger than::Finitely generated subnormal subgroup]]
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{| class="sortable" border="1"
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
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|-
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| [[Stronger than::Finitely generated subnormal subgroup]] || || || || {{intermediate notions short|finitely generated subnormal subgroup|finite subnormal subgroup}}
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|-
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| [[Stronger than::Subnormal subgroup]] || || || || {{intermediate notions short|subnormal subgroup|finite subnormal subgroup}}
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|}
  
 
===Related properties===
 
===Related properties===
  
* [[Stronger than::Subnormal subgroup of finite index]]
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* [[Subnormal subgroup of finite index]]
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==Facts==
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* [[Join of two finite subnormal subgroups is subnormal]]

Latest revision as of 22:33, 12 May 2010

This article describes a property that arises as the conjunction of a subgroup property: subnormal subgroup with a group property (itself viewed as a subgroup property): finite group
View a complete list of such conjunctions

Definition

A subgroup of a group is termed a finite subnormal subgroup if it is finite as a group and subnormal as a subgroup.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finite characteristic subgroup the subgroup is both a finite group and a characteristic subgroup (characteristic implies subnormal, via normal) use normal not implies characteristic (finite examples) Finite normal subgroup|FULL LIST, MORE INFO
Finite normal subgroup the subgroup is both a finite group and a normal subgroup (normal implies subnormal) normality is not transitive (finite examples) |FULL LIST, MORE INFO
Subnormal subgroup of finite group subnormal subgroup where the whole group is a finite group |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Finitely generated subnormal subgroup |FULL LIST, MORE INFO
Subnormal subgroup Right-transitively fixed-depth subnormal subgroup|FULL LIST, MORE INFO

Related properties

Facts