Join-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
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions


BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article is about a general term. A list of important particular cases (instances) is available at Category:Join-closed subgroup properties

History

This term is local to the wiki. To learn more about why this name was chosen for the term, and how it does not conflict with existing choice of terminology, refer the talk page

Definition

Symbol-free definition

A subgroup property p is termed join-closed if the join of a nonempty (but otherwise arbitrary, possibly infinite) collection of subgroups, each with property p, also has property p.

Definition with symbols

A subgroup property p is termed join-closed if given a group G, a nonempty indexing set I, and a collection of subgroups H_i for i \in I, such that each H_i satisfies p, the join, i.e. the subgroup generated by the H_is, also satisfies p.

Relation with other metaproperties

Stronger metaproperties

Weaker metaproperties