Normal-subhomomorph-containing subgroup

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Definition

A normal subgroup $N$ of a group $G$ is termed a normal-subhomomorph-containing subgroup if, for any subgroup $H$ of $N$ and any homomorphism of groups $\varphi:H \to G$ such that $\varphi(H)$ is a normal subgroup of $G$ , $\varphi(H)$ is contained in $N$ .