Commutator-realizable group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

A group is termed commutator-realizable if it can be realized as the commutator subgroup of some group.

Relation with other properties

Stronger properties

• Perfect group
• Cyclic group

Facts

• Characteristically metacyclic and commutator-realizable implies abelian

Metaproperties

Characteristic quotients

This group property is characteristic quotient-closed: the quotient group by any characteristic subgroup, of a group with this property, also has this property
View other characteristic quotient-closed group properties

If $G$ is commutator-realizable, and $H$ is a characteristic subgroup of $G$, $G/H$ is also a commutator-realizable group.