Characteristic subgroup of abelian group
This article describes a property that arises as the conjunction of a subgroup property: characteristic subgroup with a group property imposed on the ambient group: abelian group
View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup
Definition
A subgroup of a group is termed a characteristic subgroup of abelian group if is a characteristic subgroup of and is an abelian group.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| fully invariant subgroup of abelian group | the subgroup is a fully invariant subgroup (i.e., invariant under all endomorphisms) and the whole group is an abelian group. | fully invariant implies characteristic | characteristic not implies fully invariant in finite abelian group | |FULL LIST, MORE INFO |
| verbal subgroup of abelian group | (via fully invariant) | (via fully invariant) | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| abelian characteristic subgroup | the subgroup is characteristic and abelian as a group in its own right; the whole group need not be abelian. | Follows from abelianness is subgroup-closed | the trivial subgroup in any nontrivial group. | |FULL LIST, MORE INFO |
| characteristic central subgroup | characteristic subgroup that is contained in the center | obvious | trivial subgroup in non-abelian group | |FULL LIST, MORE INFO |
| subgroup of abelian group | the whole group is abelian; no restriction on the subgroup. | obvious | any non-characteristic subgroup of an abelian group, e.g., a cyclic subgroup in an elementary abelian group on two or more generators. | |FULL LIST, MORE INFO |
| characteristic subgroup of center | the subgroup is contained in the center and is characteristic as a subgroup of the center | trivial subgroup in non-abelian group | |FULL LIST, MORE INFO | |
| powering-invariant subgroup and also powering-invariant subgroup of abelian group | if the group | characteristic subgroup of abelian group implies powering-invariant | (any non-characteristic subgroup of a finite abelian group) | |FULL LIST, MORE INFO |
| characteristic subgroup of nilpotent group | the subgroup is characteristic and the whole group is a nilpotent group | follows from abelian implies nilpotent | follows from nilpotent not implies abelian (we can use the whole group as the subgroup of itself) | |FULL LIST, MORE INFO |