Finite verbal subgroup: Difference between revisions
(Created page with "{{subgroup property conjunction|finite subgroup|verbal subgroup}} ==Definition== A subgroup <math>H</math> of a group <math>G</matH> is termed a '''finite verbal sub...") |
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| [[Stronger than::verbal subgroup of finite type]] || || || || {{intermediate notions short|verbal subgroup of finite type|finite verbal subgroup}} | | [[Stronger than::verbal subgroup of finite type]] || || || || {{intermediate notions short|verbal subgroup of finite type|finite verbal subgroup}} | ||
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| [[Stronger than::finite fully invariant subgroup]] || finite and a [[fully invariant subgroup]]: invariant under all [[endomorphism]]s || || || {{intermediate notions short|finite fully invariant subgroup|finite verbal subgroup}} | |||
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| [[Stronger than::finite characteristic subgroup]] || finite and a [[characteristic subgroup]]: invariant under all [[automorphism]]s || || || {{intermediate notions short|finite characteristic subgroup|finite verbal subgroup}} | |||
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| [[Stronger than::finite normal subgroup]] || finite and a [[normal subgroup]]: invariant under all [[inner automorphism]]s || || || {{intermediate notions short|finite normal subgroup|finite verbal subgroup}} | |||
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Latest revision as of 20:30, 17 July 2013
This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: finite subgroup and verbal subgroup
View other subgroup property conjunctions | view all subgroup properties
Definition
A subgroup of a group is termed a finite verbal subgroup if the following equivalent conditions are satisfied:
- is a finite group and is a verbal subgroup of .
- is a finite group and is a verbal subgroup of finite type in .
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| verbal subgroup of finite group | the whole group is finite | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| verbal subgroup of finite type | |FULL LIST, MORE INFO | |||
| finite fully invariant subgroup | finite and a fully invariant subgroup: invariant under all endomorphisms | |FULL LIST, MORE INFO | ||
| finite characteristic subgroup | finite and a characteristic subgroup: invariant under all automorphisms | |FULL LIST, MORE INFO | ||
| finite normal subgroup | finite and a normal subgroup: invariant under all inner automorphisms | |FULL LIST, MORE INFO |