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 subgrou ...

H

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 ...

G

is termed malnormal if for any

x

G

but not in

H

H

and

xHx^

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 s ...

. 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 G ...

that is also malnormal must be one of these.. *Every malnormal subgroup is a special type of

C-group The C-Group culture is an archaeological culture found in Lower Nubia, which dates from ca. 2400 BCE to ca. 1550 BCE. It was named by George A. Reisner. With no central site and no written evidence about what these people called themselves, Re ...

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