3-subnormal subgroup
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
Definition
Symbol-free definition
A subgroup of a group is termed a 3-subnormal subgroup if it satisfies the following equivalent conditions:
- It is a subnormal subgroup and its subnormal depth is at most three.
- It is a 2-subnormal subgroup of a normal subgroup.
- It is a 2-subnormal subgroup in its normal closure.
- It is a normal subgroup of a 2-subnormal subgroup.
Relation with other properties
Stronger properties
- Normal subgroup
- 2-subnormal subgroup
- Commutator of a 2-subnormal subgroup and a subset: For full proof, refer: Commutator of a 2-subnormal subgroup and a subset implies 3-subnormal