C-normal subgroup

In mathematics, in the field of group theory, a subgroup ${\displaystyle H}$ of a group ${\displaystyle G}$ is called c-normal if there is a normal subgroup ${\displaystyle T}$ of ${\displaystyle G}$ such that ${\displaystyle HT=G}$ and the intersection of ${\displaystyle H}$ and ${\displaystyle T}$ lies inside the normal core of ${\displaystyle H}$.

For a weakly c-normal subgroup, we only require ${\displaystyle T}$ to be subnormal.

Here are some facts about c-normal subgroups:

• Every normal subgroup is c-normal
• Every retract is c-normal
• Every c-normal subgroup is weakly c-normal