C-normal subgroup

History
Wang introduced the notion of c-normal subgroup in his paper  C -Normality of Groups and Its Properties.

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 $$HT = G$$.

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
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.)