Perfect normal subgroup

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 $$[H,H] = [G,H] = H$$.

Stronger properties

 * Weaker than::Perfect characteristic subgroup
 * Weaker than::Perfect direct factor

Weaker properties

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