C-normal subgroup

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

For a weakly c-normal subgroup, we only require [math]\displaystyle{ T }[/math] 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

References

• Y. Wang, c-normality of groups and its properties, Journal of Algebra, Vol. 180 (1996), 954-965