2-subnormal subgroup

Comment
A 2-subnormal subgroup $$H$$ has a unique fastest ascending subnormal series $$H \le K \le G$$, where $$K$$ is the normal core of $$N_G(H)$$. It also has a unique fastest descending subnormal series $$G \ge L \ge H$$, where $$L$$ is the normal closure of $$H$$ in $$G$$. While subnormal subgroups of larger depth also have unique fastest descending subnormal series, they do not in general possess unique fastest ascending subnormal series.

Formalisms
A subgroup $$H$$ is 2-subnormal in a group $$G$$ if it satisfies the following first-order sentence:

$$\forall g \in G, \forall x,y \in H, gxg^{-1}ygx^{-1}g^{-1} \in H$$

Metaproperties
For more details of these metaproperties:

A 2-subnormal subgroup of a 2-subnormal subgroup is not necessarily 2-subnormal.

Every group is 2-subnormal as a subgroup of itself, and further, the trivial subgroup is 2-subnormal in any group.

This follows from the fact that every group is normal in itself and the trivial subgroup is also normal in every group.

An arbitrary intersection of 2-subnormal subgroups is 2-subnormal.

An arbitrary join of 2-subnormal subgroups need not be 2-subnormal. In fact, even a join of two 2-subnormal subgroups need not be 2-subnormal.

A join of 2-subnormal subgroups that are conjugate to each other is again 2-subnormal.

If $$H$$ is a 2-subnormal subgroup of $$G$$, then $$H$$ is also 2-subnormal in any intermediate subgroup $$K$$ of $$G$$.

If $$H$$ is a 2-subnormal subgroup of $$G$$ and $$K$$ is any subgroup of $$G$$, then $$H \cap K$$ is 2-subnormal in $$K$$.

If $$f:G \to K$$ is a surjective homomorphism of groups, and $$H$$ is 2-subnormal in $$G$$, then $$f(H)$$ is 2-subnormal in $$K$$.

If $$H \le G$$ and $$K, L$$ are two intermediate subgroups containing $$H$$, it may happen that $$H$$ is 2-subnormal in $$K$$ as well as in $$L$$, but is not 2-subnormal in $$\langle K, L \rangle$$.

Effect of property operators
For more information on these operators:

The right transiter of the property of being 2-subnormal is termed the property of being right-transitively 2-subnormal. A subgroup $$H$$ of a group $$G$$ is termed right-transitively 2-subnormal if any 2-subnormal subgroup $$K$$ of $$H$$ is 2-subnormal in $$G$$.

Some subgroup properties stronger than being right-transitively 2-subnormal include: base of a wreath product, transitively normal subgroup, and normal subgroup that is also a T-group (for instance, an Abelian normal subgroup).

The left transiter of the property of being 2-subnormal is termed the property of being left-transitively 2-subnormal. A subgroup $$H$$ of a group $$G$$ is termed left-transitively 2-subnormal if whenever $$G$$ is embedded as a 2-subnormal subgroup of some group $$K$$, $$H$$ is also 2-subnormal in $$K$$.

Any characteristic subgroup is left-transitively 2-subnormal, because the left transiter of normal is characteristic.

A join-transitively 2-subnormal subgroup is a subgroup whose join with any 2-subnormal subgroup is 2-subnormal. Any normal subgroup is join-transitively 2-subnormal.