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 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 G, is a normal subgroup of G, called the "Frobenius kernel", and G is the semi-direct product of H and N (Frobenius' theorem).