Malnormal subgroup explained
of a
group
is termed
malnormal if for any
in
but not in
,
and
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 anyelement 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]
Notes and References
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