Abelian characteristic subgroup: Difference between revisions
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* [[Second half of lower central series of nilpotent group comprises abelian groups]]: In particular, this means that for a group <math>G</math> of nilpotency class <math>c</math>, all the subgroups <math>\gamma_k(G), k \ge (c + 1)/2</math> are abelian characteristic subgroups. | * [[Second half of lower central series of nilpotent group comprises abelian groups]]: In particular, this means that for a group <math>G</math> of nilpotency class <math>c</math>, all the subgroups <math>\gamma_k(G), k \ge (c + 1)/2</math> are abelian characteristic subgroups. | ||
* The [[center]] of any group is an abelian characteristic subgroup. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 17:28, 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
Facts
- Second half of lower central series of nilpotent group comprises abelian groups: In particular, this means that for a group of nilpotency class , all the subgroups are abelian characteristic subgroups.
- The center of any group is an abelian characteristic subgroup.
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 |