Perfect normal subgroup

This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): perfect group
View a complete list of such conjunctions

This article describes a property that arises as the conjunction of a subgroup property: permutable subgroup with a group property (itself viewed as a subgroup property): perfect group
View a complete list of such conjunctions

Definition

Symbol-free definition

A subgroup of a group is termed a perfect normal subgroup if it satisfies the following equivalent conditions:

• It is perfect as a group and normal as a subgroup.
• It is perfect as a group and permutable as a subgroup.

Definition with symbols

A subgroup $H$ of a group $G$ is termed a perfect normal subgroup if $[H,H]=[G,H]=H$.

Relation with other properties

Stronger properties

• Perfect characteristic subgroup
• Perfect direct factor

Weaker properties

• Subgroup whose commutator subgroup equals its intersection with whole commutator subgroup
• Subgroup whose focal subgroup equals its commutator subgroup
• Subgroup whose focal subgroup equals its intersection with the commutator subgroup
• Perfect 2-subnormal subgroup
• Perfect subnormal subgroup
• Normal subgroup contained in the perfect core