Join-closed subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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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 $H_i$s, also satisfies $p$.

Relation with other metaproperties

Stronger metaproperties

• Strongly join-closed subgroup property: This is a subgroup property that is both join-closed and trivially true
• Upward-closed subgroup property

Weaker metaproperties

• Finite-join-closed subgroup property
• Conjugate-join-closed subgroup property
• Normal closure-closed subgroup property
• Characteristic closure-closed subgroup property