2-subnormality is not transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., 2-subnormal subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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Statement

A 2-subnormal subgroup of a 2-subnormal subgroup need not be 2-subnormal.

Related facts

Failures of transitivity

• Normality is not transitive
• There exist subgroups of arbitrarily large subnormal depth
• Descendant not implies subnormal
• Ascendant not implies subnormal

Left-transitively 2-subnormal subgroup

Further information: Left-transitively 2-subnormal subgroup

A subgroup $H$ of a group $G$ is termed left-transitively 2-subnormal in $G$ if whenever $G$ is 2-subnormal in some group $K$, so is $H$. Since any characteristic subgroup of a normal subgroup is normal, every characteristic subgroup is left-transitively 2-subnormal.

Right-transitively 2-subnormal subgroup

Further information: Right-transitively 2-subnormal subgroup

A subgroup $H$ of a group $G$ is termed right-transitively 2-subnormal in $G$ if whenever $K$ is a 2-subnormal subgroup of $H$, $K$ is 2-subnormal in $G$. Any transitively normal subgroup, as well as any base of a wreath product, is right-transitively 2-subnormal.

Facts used

1. There exist subgroups of arbitrarily large subnormal depth

Proof

The proof follows directly from fact (1): if every 2-subnormal subgroup of a 2-subnormal subgroup were 2-subnormal, then every subnormal subgroup would be 2-subnormal, and we would not get subgroups of arbitrarily large subnormal depth.