# ACU-closed subgroup property

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 metaproperty
VIEW 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 $p$ 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 $p$.