ACU-closed group property

From Groupprops
Jump to: navigation, search
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.

Relation with other metaproperties

Stronger metaproperties