Malnormal subgroup

In mathematics, in the field of group theory, a subgroup H {\displaystyle H} of a group G {\displaystyle G} is termed malnormal if for any x {\displaystyle x} in G {\displaystyle G} but not in H {\displaystyle H} , H {\displaystyle H} and x H x − 1 {\displaystyle xHx^{-1}} intersect only in the identity element. Some facts about malnormality: An 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 (Frobenius' theorem).