Free factor

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 $$G = H * K$$.

Stronger properties

 * Free root

Weaker properties

 * Stronger than::Regular retract
 * Stronger than::Retract
 * Self-normalizing subgroup if nontrivial:

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.