Malnormal subgroup

In mathematics, in the field of group theory, a subgroup ${\displaystyle H}$ of a group ${\displaystyle G}$ is termed malnormal if for any ${\displaystyle x}$ in ${\displaystyle G}$ but not in ${\displaystyle H}$, ${\displaystyle H}$ and ${\displaystyle xHx^{-1}}$ intersect only 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 H, is a normal subgroup of G, called the "Frobenius kernel", and G is the semidirect product of H and N (Frobenius' theorem).^[5]