Paranormal subgroup

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title: "Paranormal 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/Paranormal_subgroup" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it within that subgroup.

In symbols, H is paranormal in G if given any g in G, the subgroup K generated by H and H^g is also equal to H^K. Equivalently, a subgroup is paranormal if its weak closure and normal closure coincide in all intermediate subgroups.

Here are some facts relating paranormality to other subgroup properties:

Every pronormal subgroup, and hence, every normal subgroup and every abnormal subgroup, is paranormal.
Every paranormal subgroup is a polynormal subgroup.
In finite solvable groups, every polynormal subgroup is paranormal.

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