Abelian-completed subgroup: Difference between revisions

Revision as of 12:36, 21 February 2007

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Definition

Symbol-free definition

A subgroup of a group is termed Abelian-completed if there is an Abelian subgroup such that their product is the whole group.

Definition with symbols

A subgroup $H$ of a group $G$ is termed Abelian-completed if there is an Abelian subgroup $A$ such that $H A = G$ .

Relation with other properties

Stronger properties

Cocentral subgroup

Metaproperties

Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group
View other upward-closed subgroup properties

If $H$ and $A$ generate $G$ , then so do $K$ and $A$ for any $K$ containing $H$ . Hence, the property of being Abelian-completed is upward-closed.

@@ Line 9: / Line 9: @@
 ===Definition with symbols===
-A [[subgroup]] &lt;math&gt;H&lt;/math&gt; of a [[group]] &lt;math&gt;G&lt;/math&gt; is termed '''Abelian-completed''' if there is an [[Abelian group|Abelian]] subgroup &lt;math&gt;A&lt;/math&gt; such that &lt;math&gt;HA = G&lt;/math&gt;.
+A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''Abelian-completed''' if there is an [[Abelian group|Abelian]] subgroup <math>A</math> such that <math>HA = G</math>.
 ==Relation with other properties==
@@ Line 21: / Line 21: @@
 {{upward-closed}}
-If &lt;math&gt;H&lt;/math&gt; and &lt;math&gt;A&lt;/math&gt; generate &lt;math&gt;G&lt;/math&gt;, then so do &lt;math&gt;K&lt;/math&gt; and &lt;math&gt;A&lt;/math&gt; for any &lt;math&gt;K&lt;/math&gt; containing &lt;math&gt;H&lt;/math&gt;. Hence, the property of being Abelian-completed is upward-closed.
+If <math>H</math> and <math>A</math> generate <math>G</math>, then so do <math>K</math> and <math>A</math> for any <math>K</math> containing <math>H</math>. Hence, the property of being Abelian-completed is upward-closed.