Malnormal subgroup

In group theory, a subgroup ${\displaystyle H}$ of a group ${\displaystyle G}$ is termed malnormal if for any group element ${\displaystyle x}$ not in ${\displaystyle H}$, ${\displaystyle H}$ and ${\displaystyle xHx^{-1}}$ intersect only in the identity element.

Some facts about malnormality:

• The 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 ${\displaystyle G}$ is finite, a malnormal subgroup ${\displaystyle H}$ distinct from 1 and ${\displaystyle G}$ is called a "Frobenius complement". The set ${\displaystyle N}$ of elements of ${\displaystyle G}$ which are either equal to 1 or non-conjugate to any element of ${\displaystyle H}$ is a normal subgroup of ${\displaystyle G}$, called the "Frobenius kernel", and ${\displaystyle G}$ is the semidirect product of ${\displaystyle H}$ and ${\displaystyle N}$.

Notes

References

This group theory–related article is a stub. You can help Wikipedia by adding missing information.

• v
• t
• e