Abelian characteristic subgroup: Difference between revisions
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| [[Stronger than::abelian normal subgroup]] || abelian and a [[normal subgroup]] -- invariant under all [[inner automorphism]]s || follows from [[characteristic implies normal]] || follows from [[normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property]] || {{intermediate notions short|abelian normal subgroup|abelian characteristic subgroup}} | | [[Stronger than::abelian normal subgroup]] || abelian and a [[normal subgroup]] -- invariant under all [[inner automorphism]]s || follows from [[characteristic implies normal]] || follows from [[normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property]] || {{intermediate notions short|abelian normal subgroup|abelian characteristic subgroup}} | ||
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| [[Stronger than::abelian subnormal subgroup]] || abelian and a [[subnormal subgroup]] || || || {{intermediate notions short|abelian subnormal subgroup|abelian characteristic subgroup}} | |||
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| [[Stronger than::class two characteristic subgroup]] || characteristic subgroup that is also a [[group of nilpotency class two]] || || || {{intermediate notions short|class two characteristic subgroup|abelian characteristic subgroup}} | | [[Stronger than::class two characteristic subgroup]] || characteristic subgroup that is also a [[group of nilpotency class two]] || || || {{intermediate notions short|class two characteristic subgroup|abelian characteristic subgroup}} | ||
Revision as of 17:23, 12 August 2013
This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property (itself viewed as a subgroup property): Abelian group
View a complete list of such conjunctions
Definition
Symbol-free definition
A subgroup of a group is termed an Abelian characteristic subgroup if it is Abelian as a group and characteristic as a subgroup.
Examples
VIEW: subgroups satisfying this property | subgroups dissatisfying property characteristic subgroup | subgroups dissatisfying property abelian group
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| cyclic characteristic subgroup | cyclic group and a characteristic subgroup of the whole group | |FULL LIST, MORE INFO | ||
| characteristic central subgroup | central subgroup (i.e., contained in the center) and a characteristic subgroup of the whole group | |FULL LIST, MORE INFO | ||
| abelian critical subgroup | |FULL LIST, MORE INFO | |||
| characteristic subgroup of center | characteristic subgroup of the center | |FULL LIST, MORE INFO | ||
| characteristic subgroup of abelian group | the whole group is an abelian group and the subgroup is characteristic | |FULL LIST, MORE INFO | ||
| maximal among abelian characteristic subgroups | |FULL LIST, MORE INFO | |||
| abelian fully invariant subgroup | abelian and a fully invariant subgroup -- invariant under all endomorphisms | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| abelian normal subgroup | abelian and a normal subgroup -- invariant under all inner automorphisms | follows from characteristic implies normal | follows from normal not implies characteristic in the collection of all groups satisfying a nontrivial finite direct product-closed group property | |FULL LIST, MORE INFO |
| abelian subnormal subgroup | abelian and a subnormal subgroup | |FULL LIST, MORE INFO | ||
| class two characteristic subgroup | characteristic subgroup that is also a group of nilpotency class two | |FULL LIST, MORE INFO | ||
| nilpotent characteristic subgroup | characteristic subgroup that is also a nilpotent group | follows from abelian implies nilpotent | follows from nilpotent not implies abelian | |FULL LIST, MORE INFO |
| solvable characteristic subgroup | characteristic subgroup that is also a solvable group | (via nilpotent) | (via nilpotent) | |FULL LIST, MORE INFO |