Malnormal subgroup

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
• Malnormality is transitive, that is, a malnormal subgroup of a malnormal subgroup is malnormal.{{citation
• 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