In
mathematics, 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, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
is termed malnormal if for any
in
but not in
,
and
intersect in the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
.
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).
[.]
References
{{DEFAULTSORT:Malnormal Subgroup
Subgroup properties