# ACU-closed group property

From Groupprops

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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 is termed **ACU-closed** if, for any group , any nonempty totally ordered set , and any ascending chain of subgroups of indexed by ordinals such that for , the subgroup:

also satisfies property .