Malnormal subgroup

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In mathematics, in the field of group theory, a subgroup of a group is termed malnormal if for any in but not in , and intersect in the identity element.[1]

Some facts about malnormality:

  • An intersection of malnormal subgroups is malnormal.[2]
  • Malnormality is transitive, that is, a malnormal subgroup of a malnormal subgroup is malnormal.[3]
  • The trivial subgroup and the whole group are malnormal subgroups. A normal subgroup that is also malnormal must be one of these.[4]
  • 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".[4] The set N of elements of G which are, either equal to 1, or non-conjugate to any element of G, is a normal subgroup of G, called the "Frobenius kernel", and G is the semi-direct product of H and N (Frobenius' theorem).[5]


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