Malnormal subgroup
In mathematics, in the field of group theory, a subgroup [math]\displaystyle{ H }[/math] of a group [math]\displaystyle{ G }[/math] is termed malnormal if for any [math]\displaystyle{ x }[/math] in [math]\displaystyle{ G }[/math] but not in [math]\displaystyle{ H }[/math], [math]\displaystyle{ H }[/math] and [math]\displaystyle{ xHx^{-1} }[/math] 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 H, 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
- ↑ Lyndon, Roger C.; Schupp, Paul E. (2001), Combinatorial Group Theory, Springer, p. 203, ISBN 9783540411581, https://books.google.com/books?id=aiPVBygHi_oC&pg=PA203.
- ↑ 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.
- ↑ 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.
- ↑ 4.0 4.1 de la Harpe, Pierre; Weber, Claude (2011), Malnormal subgroups and Frobenius groups: basics and examples, Bibcode: 2011arXiv1104.3065D.
- ↑ Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967, pp. 133–139.
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