Subgroup of abelian group

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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) 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) Central subgroup|FULL LIST, MORE INFO
normal subgroup abelian implies every subgroup is normal any non-abelian group as a subgroup of itself. Central subgroup, Normal subgroup of nilpotent group|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) Central subgroup|FULL LIST, MORE INFO