Transitively normal subgroup

---

title: "Transitively normal subgroup" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["subgroup-properties"] description: "Property of a subgroup in mathematics" topic_path: "general/subgroup-properties" source: "https://en.wikipedia.org/wiki/Transitively_normal_subgroup" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Property of a subgroup in mathematics ::

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.

References

References

1. "On the influence of transitively normal subgroups on the structure of some infinite groups".

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::