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]

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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 $H$.

Weaker properties

• C-subnormal 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.
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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.)

References

• C-Normality of Groups and its properties by Wang, Journal of Algebra, Volume 180, Number 3, 1 March 1996, pp. 954-965(12)