Subgroup of abelian group
This article describes a property that arises as the conjunction of a subgroup property: 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 subgroup of abelian group if the ambient group is an abelian group.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| subgroup of cyclic group | the whole group is cyclic | |FULL LIST, MORE INFO |
Weaker properties
Properties weaker than this are properties true for all subgroups of abelian groups.
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| central subgroup | contained in the center | |FULL LIST, MORE INFO | ||
| central factor | product with centralizer is whole group | (via central subgroup) | (via central subgroup) | |FULL LIST, MORE INFO |
| transitively normal subgroup | every normal subgroup of it is normal in the whole group. | (via central factor) | (via central factor) | |FULL LIST, MORE INFO |
| normal subgroup | abelian implies every subgroup is normal | any non-abelian group as a subgroup of itself. | |FULL LIST, MORE INFO | |
| abelian normal subgroup | the subgroup is abelian as a group and is normal in the whole group. | (via central subgroup) | (via central subgroup) | |FULL LIST, MORE INFO |