# ACU-closed subgroup property

From Groupprops

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property

View a complete list of subgroup metaproperties

View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metapropertyVIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

## Definition

### 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 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 .