Intermediately subnormal-to-normal subgroup

From Groupprops

Jump to: navigation, search
This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof.
View a complete list of subgroup properties|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | | Subgroup metaproperty satisfactions | | |
RANDOM TIP:The metaproperties section lists important facts about the subgroup property, and addresses many of the natural questions that arise about it. It has links to proofs.


Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this | Survey articles about this
Learn more about terminology local to the wiki | view a complete list of such terminology

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 all properties obtained by applying the intermediately operator

Relation with other properties

Stronger properties

Weaker properties

Metaproperties

Intermediate subgroup condition

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
View all subgroup properties satisfying the intermediate subgroup condition|View facts related to the intermediate subgroup condition

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

Facts about Intermediately subnormal-to-normal subgroupRDF feed
Defining ingredientSubnormal subgroup  +, Normal subgroup  +, 2-subnormal subgroup  +, Intermediately operator  +, and Subnormal-to-normal subgroup  +
Stronger thanSubnormal-to-normal subgroup  +, and Subgroup with self-normalizing normalizer  +
Weaker thanNormal subgroup  +, Abnormal subgroup  +, Weakly abnormal subgroup  +, Pronormal subgroup  +, Weakly pronormal subgroup  +, Paranormal subgroup  +, Polynormal subgroup  +, Intermediately normal-to-characteristic subgroup  +, and Self-normalizing subgroup  +
Retrieved from "http://groupprops.subwiki.org/wiki/Intermediately_subnormal-to-normal_subgroup"
Personal tools