Normal-homomorph-containing subgroup

Definition
A subgroup $$N$$ of a group $$G$$ is termed normal-homomorph-containing if $$N$$ is a defining ingredient::normal subgroup of $$G$$, and for any homomorphism $$\varphi:N \to G$$ such that $$\varphi(N)$$ is also a normal subgroup of $$G$$, we have $$\varphi(N) \le N$$.

Stronger properties

 * Weaker than::Homomorph-containing subgroup: Also related:
 * Weaker than::Subhomomorph-containing subgroup
 * Weaker than::Order-containing subgroup
 * Weaker than::Variety-containing subgroup
 * Weaker than::Normal Sylow subgroup
 * Weaker than::Normal Hall subgroup
 * Weaker than::Normal-subhomomorph-containing subgroup

Weaker properties

 * Stronger than::Weakly normal-homomorph-containing subgroup
 * Stronger than::Strictly characteristic subgroup: Also related:
 * Stronger than::Characteristic subgroup
 * Stronger than::Normal subgroup
 * Stronger than::Normal-isomorph-containing subgroup