Malnormal subgroup

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

References

• Feit, Walter (1967). Characters of Finite Groups. New York: W. A. Benjamin. MR 0219636.

• Gildenhuys, D.; Kharlampovich, O.; Myasnikov, A. (1995). "CSA-groups and separated free constructions". Bulletin of the Australian Mathematical Society. 52 (1): 63–84. doi:10.1017/S0004972700014453. MR 1344261.

• de la Harpe, Pierre; Weber, Claude (2014). "Malnormal subgroups and Frobenius groups: basics and examples". Confluentes Mathematici. 6 (1): 65–76. arXiv:1104.3065. doi:10.5802/cml.13.

• Karrass, A.; Solitar, D. (1971). "The free product of two groups with a malnormal amalgamated subgroup". Canadian Journal of Mathematics. 23 (5): 933–959. doi:10.4153/cjm-1971-102-8. MR 0314992.

• Lyndon, Roger C.; Schupp, Paul E. (2001) [1977]. Combinatorial Group Theory. Classics in Mathematics. New York: Springer. ISBN 978-3-540-41158-1.