# C-normal subgroup

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.
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## 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 $H$ of a group $G$ is termed c-normal if there is a normal subgroup $T$ such that the intersection of $H$ and $T$ lies inside the normal core of $H$, and such that $HT = G$.

## Relation with other properties

### Stronger properties

• Normal subgroup: we can take $T$ to be the whole group $G$.
• Retract: We can take $T$ to be a normal complement to $H$.

## 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 c-normal in $G$, with $T$ being a normal subgroup that shows it, then $H$ is also c-normal in any intermediate subgroup $K$, and further, the normal subgroup that does the trick is $T \cap K$. This follows because: $H(T \cap K) = HT \cap K = G \cap K = K$

(the first step is the famous modular property of groups and can be proved easily.)