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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 {\displaystyle H} is a transitively normal subgroup of G {\displaystyle G} if for every K {\displaystyle K} normal in H {\displaystyle H} , we have that K {\displaystyle K} is normal in G {\displaystyle 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: