Free factor

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]

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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 $H$ of a group $G$ is termed a free factor if there is a subgroup $K$ of $G$ such that $G$ such that $G=H*K$.

Relation with other properties

Stronger properties

• Free root

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 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

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.