Malnormal subgroup

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title: "Malnormal 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/Malnormal_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 group G is termed malnormal if for any x in G but not in H, H and xHx^{-1} intersect only in the identity element.

Some facts about malnormality:

  • An intersection of malnormal subgroups is malnormal.{{citation | last1 = Gildenhuys | first1 = D. | last2 = Kharlampovich | first2 = O. | last3 = Myasnikov | first3 = A. | arxiv = math/9605203 | doi = 10.1017/S0004972700014453 | issue = 1 | journal = Bulletin of the Australian Mathematical Society | mr = 1344261 | pages = 63–84 | title = CSA-groups and separated free constructions | volume = 52 | year = 1995}}.
  • Malnormality is transitive, that is, a malnormal subgroup of a malnormal subgroup is malnormal.{{citation | last1 = Karrass | first1 = A. | last2 = Solitar | first2 = D. | journal = Canadian Journal of Mathematics | mr = 0314992 | pages = 933–959 | title = The free product of two groups with a malnormal amalgamated subgroup | volume = 23 | year = 1971 | doi=10.4153/cjm-1971-102-8| doi-access = free
  • The trivial subgroup and the whole group are malnormal subgroups. A normal subgroup that is also malnormal must be one of these.
  • Every malnormal subgroup is a special type of C-group called a trivial intersection subgroup or TI subgroup.

When G is finite, a malnormal subgroup H distinct from 1 and G is called a "Frobenius complement". The set N of elements of G which are, either equal to 1, or non-conjugate to any element of H, is a normal subgroup of G, called the "Frobenius kernel", and G is the semidirect product of H and N (Frobenius' theorem).{{citation | last = Feit | first = Walter | authorlink = Walter Feit | mr = 0219636 | pages = 133–139 | publisher = W. A. Benjamin, Inc., New York-Amsterdam | title = Characters of finite groups | year = 1967}}.

References

References

  1. (2001). "Combinatorial Group Theory". Springer.
  2. (2011). "Malnormal subgroups and Frobenius groups: basics and examples".

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