2-subnormal implies conjugate-join-closed subnormal

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., 2-subnormal subgroup) must also satisfy the second subgroup property (i.e., conjugate-join-closed subnormal subgroup)
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Statement

A 2-subnormal subgroup of a group is a conjugate-join-closed subnormal subgroup: the join of an arbitrary number of conjugate subgroups to a 2-subnormal subgroup is subnormal.

Facts used

  1. 2-subnormality is conjugate-join-closed

Proof

The proof follows directly from fact (1), which states that an arbitrary join of conjugate 2-subnormal subgroups is 2-subnormal. Since 2-subnormal subgroups are subnormal, the conclusion follows.