Metabelian group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

History

Origin

The concept and term metabelian group was introduced by Furtwangler in 1930.

The term metabelian was earlier used for groups of nilpotence class 2, but is no longer used in that sense.

Definition

Symbol-free definition

A group is said to be metabelian if it satisfies the following equivalent conditions:

• It is solvable of solvable length 2.
• Its commutator subgroup is Abelian.
• There is a normal Abelian subgroup such that the quotient is also Abelian.

In terms of the meta operator

This property is obtained by applying the meta operator to the property: Abelian group
View other properties obtained by applying the meta operator

The property of being metabelian arises by applying the meta operator to the group property of being Abelian. Equivalently metabelian can be described as Abelian-by-Abelian, where by denotes the group extension operator.

Relation with other properties

Stronger properties

• Abelian group
• Group of nilpotence class two
• Metacyclic group

Weaker properties

• Group with nilpotent commutator subgroup
• Abelian-by-nilpotent group
• Solvable group
• Group satisfying subnormal join property: For full proof, refer: metabelian implies subnormal join property

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
View a complete list of subgroup-closed group properties

Any subgroup of a metabelian group is metabelian. This follows from the general fact that the derived series of the subgroup is contained (entry-wise) in the derived series of the whole group.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties

Any quotient of a metabelian group is metabelian. This follows from the fact that the derived series of the quotient is the quotient of the derived series of the original group.

Direct products

This group property is direct product-closed, viz., the direct product of an arbitrary (possibly infinite) family of groups each having the property, also has the property
View other direct product-closed group properties

A direct product of metabelian groups is metabelian. This follows from the fact that the derived series of the direct product is the direct product of the respective derived series.