Subnormal subgroup
title: "Subnormal subgroup" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["subgroup-properties"] topic_path: "general/subgroup-properties" source: "https://en.wikipedia.org/wiki/Subnormal_subgroup" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G.
In notation, H is k-subnormal in G if there are subgroups
:H=H_0,H_1,H_2,\ldots, H_k=G
of G such that H_i is normal in H_{i+1} for each i.
A subnormal subgroup is a subgroup that is k-subnormal for some positive integer k. Some facts about subnormal subgroups:
- A 1-subnormal subgroup is a proper normal subgroup (and vice versa).
- A finitely generated group is nilpotent if and only if each of its subgroups is subnormal.
- Every quasinormal subgroup, and, more generally, every conjugate-permutable subgroup, of a finite group is subnormal.
- Every pronormal subgroup that is also subnormal, is normal. In particular, a Sylow subgroup is subnormal if and only if it is normal.
- Every 2-subnormal subgroup is a conjugate-permutable subgroup.
The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality.
If every subnormal subgroup of G is normal in G, then G is called a T-group.
References
::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. ::