# Malnormal subgroup

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

## Definition

### Symbol-free definition

A subgroup of a group is termed malnormal if it satisfies the following equivalent conditions:

1. Its conjugate by any element outside the subgroup intersects it trivially.
2. It is a self-normalizing subgroup and is also a TI-subgroup.

### Definition with symbols

A subgroup $H$ of a group $G$ is termed malnormal if for any $g$ outside $H$, the subgroup $gHg^{-1}$ intersects $H$ trivially.

## Relation with other properties

### Oppositeness to normality

The only normal malnormal subgroup of a group is the whole group itself.

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

A malnormal subgroup of a malnormal subgroup is malnormal. That is, if $G \le H \le K$ such that $H$ is malnormal in $K$ and $G$ is malnormal in $H$, then $G$ is malnormal in $K$. The proof relies on the fact that every element in $K \setminus G$ lies either in $K \setminus H$ or $H \setminus G$, and the use of malnormality in each case. For full proof, refer: Malnormality is transitive

### Trimness

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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### Transfer condition

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 $H$ is a malnormal subgroup of $G$ and $K$ is any subgroup of $G$, then $H \cap K$ is clearly malnormal in $K$.

### Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness

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.