Normal implies join-transitively subnormal

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., normal subgroup) must also satisfy the second subgroup property (i.e., join-transitively subnormal subgroup)
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Statement

Verbal statement

1. Any normal subgroup of a group is a join-transitively subnormal subgroup.
2. The join of a normal subgroup and a subnormal subgroup is subnormal.

Statement with symbols

1. If $H$ is a normal subgroup of $G$, $H$ is also join-transitively subnormal in $G$.
2. If $H$ is a normal subgroup of $G$ and $K$ is a subnormal subgroup of $G$, $HK$ is a subnormal subgroup of $G$.

Related facts

• Join of normal and subnormal implies subnormal of same depth
• Normal implies join-transitively 2-subnormal
• 2-subnormal implies join-transitively subnormal
• Permutable and subnormal implies join-transitively subnormal

Facts used

1. Join of normal and subnormal implies subnormal of same depth

Proof

The proof is direct from fact (1).