Lesson plan

Malnormal subgroup

In group theory, a subgroup H of a group G is termed malnormal if for any group element x not in H , H and xHx^{-1} intersect only in the identity element. Some facts about malnormality: * The intersection of malnormal subgroups is malnormal. * Malnormality is transitive; that is, a malnormal subgroup of a malnormal subgroup is malnormal. * 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 .

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Article summary

In group theory, a subgroup H of a group G is termed malnormal if for any group element x not in H , H and xHx^{-1} intersect only in the identity element. Some facts about malnormality: * The intersection of malnormal subgroups is malnormal. * Malnormality is transitive; that is, a malnormal subgroup of a malnormal subgroup is malnormal. * 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 .

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  1. Summarise the article in your own words. Which single claim would you verify first, and why?
  2. Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
  3. If two sources disagreed about one of these facts, what would you need in order to trust one over the other?

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