Free factor

From Groupprops
Jump to: navigation, search
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]

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

Weaker properties

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.