2-subnormality is strongly intersection-closed

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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

  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_is is normal in the intersection of the K_is.

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_is 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.)