2-subnormal subgroup

Definition

Symbol-free definition

A subgroup of a group is termed 2-subnormal if the following equivalent conditions hold:

• There is an intermediate subgroup containing it such that the subgroup is normal in the intermediate subgroup and such that the intermediate subgroup is normal in the whole group.
• The subgroup is normal in its normal closure.

The property of being 2-subnormal is the same as the property of being subnormal of depth 2.

Definition with symbols

A subgroup $H$ of a group $G$ is termed 2-subnormal if the following equivalent conditions hold:

• There is subgroup $K$ such that $H$ is a normal subgroup of $K$ and $K$ is a normal subgroup of $G$.
• The normal closure of $H$ is a normal subgroup of $G$.

Relation with other properties

Stronger properties

• Normal subgroup: This follows directly from the definition.
• 2-hypernormalized subgroup: This is a particular case of the fact that any $k$-hypernormalized subgroup is also $k$-subnormal.

Weaker properties

• Conjugate-permutable subgroup: For full proof, refer: 2-subnormal implies conjugate-permutable
• Subnormal subgroup: This follows directly from the definition.