2-subnormality is strongly intersection-closed

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) uses::Normality is strongly intersection-closed: An arbitrary intersection of normal subgroups is normal.
 * 2) uses::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 $$G$$.)