# Free factor

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed a **free factor** if the group can be expressed as an internal free product with that subgroup as one of the factors.

### Definition with symbols

A subgroup of a group is termed a **free factor** if there is a subgroup of such that such that .

## Relation with other properties

### Stronger properties

### Weaker properties

- Regular retract
- Retract
- Self-normalizing subgroup if nontrivial:
`For full proof, refer: Free factor implies self-normalizing or trivial`

## 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

## Counterexamples it gives

### Self-normalizing subgroups that are not contranormal

A free factor is self-normalizing, but no nontrivial free factor is contranormal. This gives an example of a subgroup that is self-normalizing but not contranormal.