Conjugate-permutable subgroup

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title: "Conjugate-permutable 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/Conjugate-permutable_subgroup" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In mathematics, in the field of group theory, a conjugate-permutable subgroup is a subgroup that commutes with all its conjugate subgroups. The term was introduced by Tuval Foguel in 1997{{citation | last = Foguel | first = Tuval | doi = 10.1006/jabr.1996.6924 | issue = 1 | journal = Journal of Algebra | mr = 1444498 | pages = 235–239 | title = Conjugate-permutable subgroups | volume = 191 | year = 1997| doi-access =

Clearly, every quasinormal subgroup is conjugate-permutable.

In fact, it is true that for a finite group:

  • Every maximal conjugate-permutable subgroup is normal.
  • Every conjugate-permutable subgroup is a conjugate-permutable subgroup of every intermediate subgroup containing it.
  • Combining the above two facts, every conjugate-permutable subgroup is subnormal.

Conversely, every 2-subnormal subgroup (that is, a subgroup that is a normal subgroup of a normal subgroup) is conjugate-permutable.

References

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