Upper 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:Upper join-closed subgroup properties
Definition with symbols
A subgroup property is said to be upper join-closed if given and are intermediate subgroups of containing (indexed by a nonempty set ) and satisfies in each , we have that satisfies in the join of subgroups .
Relation with other metaproperties
- Lower-intersection upper-join closed subgroup property
- LU-join closed subgroup property
- Upward-closed subgroup property
- Izable subgroup property
Given a subgroup property that is identity-true, upper join-closed and also satisfies the intermediate subgroup condition, we can, given any subgroup of associate a unique largest subgroup containing for which satisfies in .
Properties satisfying it
Normality is an upper join-closed subgroup property, viz, if and are intermediate subgroups such that and , then .
The property of being a central factor is also upper join-closed, in fact, it is izable.