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.