Intermediately fully invariant subgroup: Difference between revisions

Revision as of 02:53, 22 May 2009

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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]
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Definition

Symbol-free definition

A subgroup of a group is termed intermediately fully invariant or intermediately fully characteristic if it is fully invariant in every intermediate subgroup of the group containing it.

Definition with symbols

A subgroup $H$ of a group $G$ is termed intermediately fully invariant or intermediately fully characteristic in $G$ if, for any intermediate subgroup $K$ of $G$ , $H$ is fully characteristic in $K$ : for any endomorphism $φ$ of $K$ , $φ (H) \leq H$ .

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity

An intermediately fully characteristic subgroup of an intermediately fully characteristic subgroup need not be intermediately fully characteristic. For full proof, refer: Intermediate full characteristicity is not transitive

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are join-closed | View variations of this property that are not join-closed
ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

An arbitrary join of intermediately fully characteristic subgroups is intermediately fully characteristic. This follows from the fact that the intermediately operator preserves the property of being closed under joins. For full proof, refer: Intermediate full characteristicity is strongly join-closed

@@ Line 6: / Line 6: @@
 ===Symbol-free definition===
-A [[subgroup]] of a [[group]] is termed '''intermediately fully characteristic''' if it is [[defining ingredient::fully characteristic subgroup|fully characteristic]] in every intermediate subgroup of the group containing it.
+A [[subgroup]] of a [[group]] is termed '''intermediately fully invariant''' or '''intermediately fully characteristic''' if it is [[defining ingredient::fully invariant subgroup|fully invariant]] in every intermediate subgroup of the group containing it.
 ===Definition with symbols===
-A subgroup <math>H</math> of a group <math>G</math> is termed '''intermediately fully characteristic''' in <math>G</math> if, for any intermediate subgroup <math>K</math> of <math>G</math>, <math>H</math> is fully characteristic in <math>K</math>: for any [[endomorphism]] <math>\varphi</math> of <math>K</math>, <math>\varphi(H) \le H</math>.
+A subgroup <math>H</math> of a group <math>G</math> is termed '''intermediately fully invariant''' or '''intermediately fully characteristic''' in <math>G</math> if, for any intermediate subgroup <math>K</math> of <math>G</math>, <math>H</math> is fully characteristic in <math>K</math>: for any [[endomorphism]] <math>\varphi</math> of <math>K</math>, <math>\varphi(H) \le H</math>.
 ==Relation with other properties==
@@ Line 22: / Line 22: @@
 ===Weaker properties===
-* [[Stronger than::Fully characteristic subgroup]]
+* [[Stronger than::Fully invariant subgroup]]
 * [[Stronger than::Intermediately characteristic subgroup]]
 * [[Stronger than::Characteristic subgroup]]