C-normal 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]
This is a variation of normality|Find other variations of normality | Read a survey article on varying normality
This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
VIEW: Definitions built on this | Facts about this: (facts closely related to C-normal subgroup, all facts related to C-normal subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a list of other standard non-basic definitions
History
This term was introduced by: Wang
Wang introduced the notion of c-normal subgroup in his paper C -Normality of Groups and Its Properties.
Definition
Symbol-free definition
A subgroup of a group is termed c-normal if there is a normal subgroup whose product with it is the whole group and whose intersection with it lies inside its normal core.
Definition with symbols
A subgroup of a group is termed c-normal if there is a normal subgroup such that the intersection of and lies inside the normal core of , and such that .
Relation with other properties
Stronger properties
- Normal subgroup: we can take to be the whole group .
- Retract: We can take to be a normal complement to .
Weaker properties
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 is c-normal in , with being a normal subgroup that shows it, then is also c-normal in any intermediate subgroup , and further, the normal subgroup that does the trick is . This follows because:
(the first step is the famous modular property of groups and can be proved easily.)
References
- C-Normality of Groups and its properties by Wang, Journal of Algebra, Volume 180, Number 3, 1 March 1996, pp. 954-965(12)