Characteristic direct factor of nilpotent group

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


 * 1) $$G$$ is a nilpotent group and $$H$$ is a characteristic direct factor of $$G$$ (i.e., $$H$$ is both a characteristic subgroup of $$G$$ and a direct factor of $$G$$).
 * 2) $$G$$ is a nilpotent group and $$H$$ is a fully invariant direct factor of $$G$$ (i.e., $$H$$ is both a fully invariant subgroup of $$G$$ and a direct factor of $$G$$). This has other equivalent formulations; see equivalence of definitions of fully invariant direct factor.

Equivalence of definitions
The equivalence follows indirectly from the fact that nontrivial subgroup of nilpotent group has nontrivial homomorphism to center.