Intermediately subnormal-to-normal subgroup: Difference between revisions

Revision as of 22:31, 1 September 2008

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 subnormal-to-normal if it satisfies the following equivalent conditions:

Whenever it is subnormal in any intermediate subgroup, then it is also normal in that intermediate subgroup.
Whenever it is 2-subnormal in any intermediate subgroup, then it is also normal in that intermediate subgroup.

Formalisms

In terms of the intermediately operator

This property is obtained by applying the intermediately operator to the property: subnormal-to-normal subgroup
View other properties obtained by applying the intermediately operator

Relation with other properties

Stronger properties

Weaker properties

Subnormal-to-normal subgroup

Metaproperties

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H$ is intermediately subnormal-to-normal in $G$ , it is also intermediately subnormal-to-normal in any intermediate subgroup $K$ .

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 ===Symbol-free definition===
-A [[subgroup]] of a [[group]] is termed '''intermediately subnormal-to-normal''' if whenever it is [[subnormal subgroup|subnormal]] in any intermediate subgroup, then it is also [[normal subgroup|normal]] in that intermediate subgroup.
+A [[subgroup]] of a [[group]] is termed '''intermediately subnormal-to-normal''' if it satisfies the following equivalent conditions:
+* Whenever it is [[defining ingredient::subnormal subgroup|subnormal]] in any intermediate subgroup, then it is also [[defining ingredient::normal subgroup|normal]] in that intermediate subgroup.
+* Whenever it is [[defining ingredient::2-subnormal subgroup|2-subnormal]] in any intermediate subgroup, then it is also [[defining ingredient::normal subgroup|normal]] in that intermediate subgroup.
 ==Formalisms==