ACU-closed subgroup property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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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 $H_{i}\leq H_{j}$ for $i<j$, the subgroup:

$\bigcup _{i\in I}H_{i}$

also satisfies property $p$.

Relation with other metaproperties

Stronger metaproperties

• Join-closed subgroup property
• Union-closed subgroup property
• Directed union-closed subgroup property