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From Groupprops

This page describes a subgroup property obtained as a composition of two fundamental subgroup properties: normal subgroup and 2-subnormal subgroup
View other such compositions|View all subgroup properties

Contents 1 Definition 1.1 Symbol-free definition 2 Relation with other properties 2.1 Stronger properties 2.2 Weaker properties

Definition

Symbol-free definition

A subgroup of a group is termed a 3-subnormal subgroup if it satisfies the following equivalent conditions:

1. It is a subnormal subgroup and its subnormal depth is at most three.
2. It is a 2-subnormal subgroup of a normal subgroup.
3. It is a 2-subnormal subgroup in its normal closure.
4. 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

Weaker properties

• Subnormal subgroup
• Conjugate-join-closed subnormal subgroup: For full proof, refer: 3-subnormal implies finite-conjugate-join-closed subnormal