ACU-closed subgroup property

Symbol-free definition
A subgroup property is termed ACU-closed (or closed under ascending chain unions) if given any ascending chain of subgroups, each of which has the property, the union of those subgroups also has the property. The ascending chain here is indexed by natural numbers.

Definition with symbols
A subgroup property $$p$$ is termed ACU-closed if, for any group $$G$$, any nonempty totally ordered set $$I$$, and any ascending chain $$H_i$$ of subgroups of $$G$$ indexed by ordinals $$i \in I$$ such that $$H_i \le H_j$$ for $$i < j$$, the subgroup:

$$\bigcup_{i \in I} H_i$$

also satisfies property $$p$$.

Stronger metaproperties

 * Weaker than::Join-closed subgroup property
 * Weaker than::Union-closed subgroup property
 * Weaker than::Directed union-closed subgroup property