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

Facts used

Normality is strongly intersection-closed: An arbitrary intersection of normal subgroups is normal.
Normality is strongly UL-intersection-closed: if $H_{i}\leq K_{i}\leq G$ 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 $G$ .)