Group with unique largest Abelian characteristic subgroup

Definition

A group $G$ is said to have unique largest Abelian characteristic subgroup if there exists an Abelian characteristic subgroup $K\leq G$ such that for every Abelian characteristic subgroup $H\leq G$, $H$ is contained in $K$.

Relation with other properties

Stronger properties

• Abelian group
• Simple group
• Characteristically simple group
• Characteristic-comparable group