Transitively normal subgroup

In mathematics, in the field of group theory, a subgroup of a group is said to be transitively normal in the group if every normal subgroup of the subgroup is also normal in the whole group. In symbols, $H$ is a transitively normal subgroup of $G$ if for every $K$ normal in $H$ , we have that $K$ is normal in $G$ .

An alternate way to characterize these subgroups is: every normal subgroup preserving automorphism of the whole group must restrict to a normal subgroup preserving automorphism of the subgroup.

Here are some facts about transitively normal subgroups:

Every normal subgroup of a transitively normal subgroup is normal.
Every direct factor, or more generally, every central factor is transitively normal. Thus, every

central subgroup is transitively normal.

A transitively normal subgroup of a transitively normal subgroup is transitively normal.
A transitively normal subgroup is normal.

Also see: Normal subgroup