Transitively normal subgroup

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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 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

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