# Transitively normal subgroup

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{{ safesubst:#invoke:Unsubst||\$N=Unreferenced |date=__DATE__ |\$B= {{#invoke:Message box|ambox}} }} 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, ${\displaystyle H}$ is a transitively normal subgroup of ${\displaystyle G}$ if for every ${\displaystyle K}$ normal in ${\displaystyle H}$, we have that ${\displaystyle K}$ is normal in ${\displaystyle G}$.

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:

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