Nilpotent subnormal subgroup

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

Definition

A subgroup of a group is termed a nilpotent subnormal subgroup if it is nilpotent as a group and subnormal as a subgroup.

Examples

Here are some examples of subgroups in basic/important groups satisfying the property:

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian subnormal subgroup	the subgroup is an abelian group and is a subnormal subgroup of the whole group			|FULL LIST, MORE INFO
abelian normal subgroup	the subgroup is an abelian group and is a normal subgroup of the whole group			|FULL LIST, MORE INFO
abelian characteristic subgroup	the subgroup is an abelian group and is a characteristic subgroup of the whole group			|FULL LIST, MORE INFO
nilpotent normal subgroup	nilpotent and a normal subgroup			|FULL LIST, MORE INFO
hereditarily normal subgroup	every subgroup of it is normal			|FULL LIST, MORE INFO
nilpotent characteristic subgroup	nilpotent and a characteristic subgroup			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
subnormal subgroup				|FULL LIST, MORE INFO
hereditarily subnormal subgroup				|FULL LIST, MORE INFO
right-transitively fixed-depth subnormal subgroup				|FULL LIST, MORE INFO
solvable subnormal subgroup				|FULL LIST, MORE INFO