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

Definition with symbols

A subgroup property $p$ is said to be upper join-closed if whenever $H \leq G$ and $K_{1}, K_{2}$ are intermediate subgroups of $G$ containing $H$ , then:

$H$ satisfies $p$ in $K_{1}$ and $H$ satisfies $p$ in $K_{2} ⟹ H$ satisfies $p$ in $< K_{1}, K_{2} >$ .

Relation with other metaproperties

Stronger 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 $H \leq G$ and $K_{1}, K_{2}$ are intermediate subgroups such that $H ◃ K_{1}$ and $H ◃ K_{2}$ , then $H ◃ < K_{1}, K_{2} >$ .

Central factor

The property of being a central factor is also upper join-closed, in fact, it is izable.