Directed union-closed group property

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

Symbol-free definition

A group property is termed directed union-closed if given any directed set of subgroups of the group, each satisfying the property, their union also satisfies the property.

Definition with symbols

A group property $p$ is termed directed union-closed if given any group $G$, any nonempty directed set $I$, and a collection of subgroups $H_{i},i\in I$ of $G$ such that $i<j\implies H_{i}\leq H_{j}$, such that each $H_{i}$ satisfies $p$, the union:

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

also satisfies $p$.

Relation with other metaproperties

Stronger metaproperties

• Join-closed group property
• Union-closed group property
• Varietal group property: For full proof, refer: Varietal implies directed union-closed

Weaker metaproperties

• ACU-closed group property