Malnormal subgroup

In mathematics, in the field of group theory, a subgroup $H$ of a group $G$ is termed malnormal if for any $x$ in $G$ but not in $H$, $H$ and $xHx^{-1}$ intersect in the identity element.^[1]

Some facts about malnormality:

• An intersection of malnormal subgroups is malnormal.^[2]
• Malnormality is transitive, that is, a malnormal subgroup of a malnormal subgroup is malnormal.^[3]
• The trivial subgroup and the whole group are malnormal subgroups. A normal subgroup that is also malnormal must be one of these.^[4]
• 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".^[4] The set N of elements of G which are, either equal to 1, or non-conjugate to any element of G, is a normal subgroup of G, called the "Frobenius kernel", and G is the semi-direct product of H and N (Frobenius' theorem).^[5]

References

1. Lyndon, Roger C.; Schupp, Paul E. (2001), Combinatorial Group Theory, Springer, p. 203, ISBN 9783540411581.
2. Gildenhuys, D.; Kharlampovich, O.; Myasnikov, A. (1995), "CSA-groups and separated free constructions", Bulletin of the Australian Mathematical Society 52 (1): 63–84, arXiv:math/9605203, doi:10.1017/S0004972700014453, MR 1344261.
3. Karrass, A.; Solitar, D. (1971), "The free product of two groups with a malnormal amalgamated subgroup", Canadian Journal of Mathematics 23: 933–959, doi:10.4153/cjm-1971-102-8, MR 0314992.
4. de la Harpe, Pierre; Weber, Claude (2011), Malnormal subgroups and Frobenius groups: basics and examples, arXiv:1104.3065.
5. Feit, Walter (1967), Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, pp. 133–139, MR 0219636.