In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in the field of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
of a
group is termed malnormal if for any
in
but not in
,
and
intersect only in the
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
.
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
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group ...
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).
[.]
References
{{DEFAULTSORT:Malnormal Subgroup
Subgroup properties