# Upper 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
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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

## Definition

### Definition with symbols

A subgroup property $p$ is said to be upper join-closed if given $H \le G$ and $K_i, i \in I$ are intermediate subgroups of $G$ containing $H$ (indexed by a nonempty set $I$) and $H$ satisfies $p$ in each $K_i$, we have that $H$ satisfies $p$ in the join of subgroups $\langle K_i \rangle_{i \in I}$.

## 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 $H \le G$ and $K_1, K_2$ are intermediate subgroups such that $H \triangleleft K_1$ and $H \triangleleft K_2$, then $H \triangleleft $.

### Central factor

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