2-subnormality is strongly intersection-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., 2-subnormal subgroup) satisfying a subgroup metaproperty (i.e., strongly intersection-closed subgroup property)
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Statement

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

Related facts

• Normality is strongly intersection-closed
• Subnormality of bounded depth is strongly intersection-closed
• Subnormality is not intersection-closed
• Normality is strongly UL-intersection-closed

Facts used

1. Normality is strongly intersection-closed: An arbitrary intersection of normal subgroups is normal.
2. Normality is strongly UL-intersection-closed: If $H_i\le K_i\le G$ are subgroups for $i\in I$ and $H_i$ is normal in $K_i$ for each $i\in I$, then the intersection of the $H_i$s is normal in the intersection of the $K_i$s.

Proof

Given: A group $G$, a collection $H_i,i\in I$ of 2-subnormal subgroups of $G$.

To prove: The intersection ${\bigcap }_{i\in I}{H}_i$ is also a 2-subnormal subgroup of $G$.

Proof: Let $K_i$ be the normal closure of $H_i$ in $G$. Thus, each $K_i$ is normal in $G$. Note that, by the definition of 2-subnormality, $H_i$ is normal in $K_i$ for each $i\in I$.

Let $H$ be the intersection of the $H_i$s and $K$ be the intersection of the $K_i$. By Fact (1), $K$ is normal in $G$. By Fact (2), $H$ is normal in $K$. Thus, $H$ is 2-subnormal in $G$.

(Note that $K$ is not necessarily the normal closure of $H$ in $G$ -- we can only say that it contains the normal closure of $H$ in $G$.)