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