Fully invariant direct factor

Definition
A subgroup of a group is termed a fully invariant direct factor if it satisfies the following equivalent conditions:


 * 1) It is both a fully invariant subgroup and a direct factor.
 * 2) It is both a defining ingredient::homomorph-containing subgroup  and a direct factor.
 * 3) It is both an defining ingredient::isomorph-containing subgroup   and a direct factor.
 * 4) It is both a defining ingredient::quotient-subisomorph-containing subgroup  and a direct factor.
 * 5) It is both a defining ingredient::normal subgroup having no nontrivial homomorphism to its quotient group  and a direct factor.

Extreme examples

 * Every group is a fully invariant direct factor in itself.
 * The trivial subgroup is a fully invariant direct factor in every group.