NE-subgroup
From Groupprops
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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Contents
Origin
The notion of NE-subgroup weas used by Yangming Li in his paper in the Journal of Group Theory.
Definition
Symbol-free definition
A subgroup of a group is termed a NE-subgroup if it equals the intersection of its normalizer and normal closure.
Definition with symbols
A subgroup of a group is termed a NE-subgroup if ∩ .
Relation with other properties
Stronger properties
Weaker properties
- Weakly normal subgroup
- Intermediately subnormal-to-normal subgroup
- Subgroup with self-normalizing normalizer
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
Since the normalizer inside an intermediate subgroup is contained inside the whole normalizer, and since the normal closure inside an intermediate subgroup is also contained inside the whole normal subgroup, we have the following: If is a NE-subgroup of and is an intermediate subgroup containing , then is a NE-subgroup of . IN other words, satisfies the intermediate subgroup condition.
References
- Finite groups with NE-subgroups by Yangming Li, Journal of Group Theory, Volume 9, Issue 1, Cover date: January 2006, Page(s): 49-58
External links
- Online version of Li's paper: Only for people who have subscribed for access