Asymptotically fixed-depth join-transitively subnormal subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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If the ambient group is a finite group, this property is equivalent to the property: subnormal subgroup
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Definition

A subgroup $H$ of a group $G$ is termed an asymptotically fixed-depth join-transitively subnormal subgroup if there exists a natural number $n$ such that for any $k \ge n$ , and any $k$ -subnormal subgroup $K$ of $G$ , the join $\langle H, K \rangle$ is also a $k$ -subnormal subgroup of $G$ .