Abelian characteristic subgroup
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
A subgroup of a group is termed an abelian characteristic subgroup if is abelian as a group (i.e., is an abelian subgroup of and also, is a characteristic subgroup of , i.e., is invariant under all automorphisms of .
Here are some examples of subgroups in basic/important groups satisfying the property:
|Group part||Subgroup part||Quotient part|
|A3 in S3||Symmetric group:S3||Cyclic group:Z3||Cyclic group:Z2|
Here are some examples of subgroups in relatively less basic/important groups satisfying the property:
Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:
- 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
|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||Characteristic central subgroup|FULL LIST, MORE INFO|
|characteristic subgroup of abelian group||the whole group is an abelian group and the subgroup is characteristic||Characteristic central subgroup, Characteristic subgroup of center|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|