# Cyclonormal subgroup

From Groupprops

This article defines a subgroup property related to (or which arises in the context of): geometric group theory

View other subgroup properties related to geometric group theory|View other terms related to geometric group theory | View all subgroup properties

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.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Cyclonormal subgroup, all facts related to Cyclonormal subgroup) |Survey articles about this | Survey articles about definitions built on this

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View a list of other standard non-basic definitions

*This is an opposite of normality*

## Definition

A subgroup of a group is termed **cyclonormal** if for any , the intersection is either the trivial group, or a cyclic subgroup.

## Relation with other properties

### Stronger properties

- Malnormal subgroup
- cyclic subgroup

## 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 transitiveABOUT 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

### 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).

View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties

### 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-closedABOUT 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