Upper join-closed subgroup property
From Groupprops
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
This article is about a general term. A list of important particular cases (instances) is available at Category:Upper join-closed subgroup properties
Contents |
Definition
Definition with symbols
A subgroup property p is said to be upper join-closed if given 
and 
are intermediate subgroups of G containing H (indexed by a nonempty set I) and H satisfies p in each Ki, we have that H satisfies p in the join of subgroups 
.
Relation with other metaproperties
Stronger metaproperties
- Lower-intersection upper-join closed subgroup property
- LU-join closed subgroup property
- Upward-closed subgroup property
- Izable subgroup property
Weaker metaproperties
Related notions
Given a subgroup property p that is identity-true, upper join-closed and also satisfies the intermediate subgroup condition, we can, given any subgroup H of G associate a unique largest subgroup M containing H for which H satisfies p in M.
Such a subgroup property is termed an izable subgroup property and the M that we get is termed the izing subgroup of H for that subgroup property.
Properties satisfying it
Normality
Normality is an upper join-closed subgroup property, viz, if and K1,K2 are intermediate subgroups such that 
and 
, then 
.
Central factor
The property of being a central factor is also upper join-closed, in fact, it is izable.
