Directed union-closed group property: Difference between revisions

Latest revision as of 17:09, 7 September 2008

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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Definition

Symbol-free definition

A group property is termed directed union-closed if given any directed set of subgroups of the group, each satisfying the property, their union also satisfies the property.

Definition with symbols

A group property $p$ is termed directed union-closed if given any group $G$ , any nonempty directed set $I$ , and a collection of subgroups $H_{i}, i \in I$ of $G$ such that $i < j ⟹ H_{i} \leq H_{j}$ , such that each $H_{i}$ satisfies $p$ , the union:

$⋃_{i \in I} H_{i}$

also satisfies $p$ .

Relation with other metaproperties

Stronger metaproperties

Join-closed group property
Union-closed group property
Varietal group property: For full proof, refer: Varietal implies directed union-closed

Weaker metaproperties

ACU-closed group property

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 ===Definition with symbols===
-A [[group property]] <math>p</math> is termed '''directed union-closed''' if given any group <math>G</math>, any directed set <math>I</math>, and a collection of subgroups <math>H_i, i \in I</math> of <math>G</math> such that <math>i < j \implies H_i \le H_j</math>, such that each <math>H_i</math> satisfies <math>p</math>, the union:
+A [[group property]] <math>p</math> is termed '''directed union-closed''' if given any group <math>G</math>, any nonempty directed set <math>I</math>, and a collection of subgroups <math>H_i, i \in I</math> of <math>G</math> such that <math>i < j \implies H_i \le H_j</math>, such that each <math>H_i</math> satisfies <math>p</math>, the union:
 <math>\bigcup_{i \in I} H_i</math>