Characteristic direct factor of abelian group

Definition
A subgroup $$H$$ of a group $$G$$ is termed a characteristic direct factor of $$G$$ if the following equivalent conditions are satisfied:


 * 1) $$G$$ is an abelian group and $$H$$ is a characteristic direct factor of $$G$$ (i.e., $$H$$ is both a characteristic subgroup of $$G$$ and a direct factor in $$G$$).
 * 2) $$G$$ is an abelian group and $$H$$ is a fully invariant direct factor of $$G$$ (i.e., $$H$$ is a fully invariant subgroup as well as a direct factor of $$G$$). See also equivalence of definitions of fully invariant direct factor for other equivalent formulations of this.