Subnormality is not finite-join-closed

Statement
A join of two subnormal subgroups of a group need not be a subnormal subgroup. In fact, it might well be a proper fact about::contranormal subgroup: its normal closure might be the whole group.

Partial truth of the opposite
A group satisfying subnormal join property is a group where a join of finitely many subnormal subgroups is subnormal. Every finite group, every Noetherian group, as well as every group satisfying ascending chain condition on subnormal subgroups satisfies the subnormal join property. For more, see ascending chain condition on subnormal subgroups implies subnormal join property.

A join-transitively subnormal subgroup is a subgroup whose join with any subnormal subgroup is again subnormal. There are many join-transitively subnormal subgroups. For instance:


 * join of normal and subnormal implies subnormal of same depth
 * 2-subnormal implies join-transitively subnormal

There are other conditions under which joins of finitely many subnormal subgroups are subnormal:


 * Subnormality is normalizing join-closed
 * Subnormality is permuting join-closed

Facts used

 * 1) uses::Join of two 3-subnormal subgroups may be proper and contranormal

Proof
The proof follows directly from fact (1), or, more concretely, from the examples used to prove fact (1). Note that since 2-subnormal implies join-transitively subnormal, getting both subgroups to be 3-subnormal is the best we can achieve in terms of subnormal depth.