This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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This article defines a subgroup property related to (or which arises in the context of): geometric group theory
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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: self-normalizing subgroup and TI-subgroup
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This is an opposite of normality
A subgroup of a group is termed malnormal if it satisfies the following equivalent conditions:
- Its conjugate by any element outside the subgroup intersects it trivially.
- It is a self-normalizing subgroup and is also a TI-subgroup.
Definition with symbols
A subgroup of a group is termed malnormal if for any outside , the subgroup intersects trivially.
Relation with other properties
- Frobenius subgroup (also called Frobenius complement) is defined as a proper nontrivial malnormal subgroup in a finite group
Oppositeness to normality
The only normal malnormal subgroup of a group is the whole group itself.
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
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A malnormal subgroup of a malnormal subgroup is malnormal. That is, if such that is malnormal in and is malnormal in , then is malnormal in . The proof relies on the fact that every element in lies either in or , and the use of malnormality in each case. For full proof, refer: Malnormality is transitive
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
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Every group is malnormal as a subgroup of itself: the condition is vacuously true because there is no element outside.
The trivial group is also always malnormal, because its intersection with anything is trivial.
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
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YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
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if is a malnormal subgroup of and is any subgroup of , then is clearly malnormal in .
YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
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An arbitrary intersection of malnormal subgroups is malnormal. This follows from the fact that any element outside the intersection must lie outside at least one of the subgroups being intersected.