ACU-closed group property

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This article defines a group metaproperty: a property that can be evaluated to true/false for any group property
View a complete list of group metaproperties

Definition

Symbol-free definition

A group property is termed ACU-closed if, whenever there is an ascending chain of subgroups in a group, each having the group property, the union of those subgroups also has the property.

Definition with symbols

A group 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 $H_{i}\leq H_{j}$ for $i<j$ , the subgroup: