Abelian subgroup

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

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Definition

A subgroup of a group is termed an abelian subgroup if it is abelian as a group.

Relation with other properties

Conjunction with other properties

• Central subgroup: An Abelian subgroup that is also a central factor.
• Abelian normal subgroup: An Abelian subgroup that is also a normal subgroup.
• Abelian characteristic subgroup: An Abelian subgroup that is also a characteristic subgroup.
• Maximal among Abelian subgroups: An Abelian subgroup that is also a self-centralizing subgroup.

Stronger properties

• Central subgroup
• Cyclic subgroup

Metaproperties

Left-hereditariness

This subgroup property is left-hereditary: any subgroup of a subgroup with this property also has this property. Hence, it is also a transitive subgroup property.

Join-closedness

This subgroup property is not join-closed, viz., it is not true that a join of subgroups with this property must have this property.
Read an article on methods to prove that a subgroup property is not join-closed

Effect of property modifiers

The join-transiter

Applying the join-transiter to this property gives: central subgroup