# ACU-closed group property

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 $H_i \le H_j$ for $i < j$, the subgroup:

$\bigcup_{i \in I} H_i$

also satisfies property $p$.