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
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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 \le 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

Weaker metaproperties