Normal-homomorph-containing subgroup

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Definition

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

Relation with other properties

Stronger properties

• Homomorph-containing subgroup: Also related:
  • Subhomomorph-containing subgroup
  • Order-containing subgroup
  • Variety-containing subgroup
  • Normal Sylow subgroup
  • Normal Hall subgroup
• Normal-subhomomorph-containing subgroup

Weaker properties

• Weakly normal-homomorph-containing subgroup
• Strictly characteristic subgroup: For proof of the implication, refer Normal-homomorph-containing implies strictly characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Strictly characteristic not implies normal-homomorph-containing. Also related:
  • Characteristic subgroup
  • Normal subgroup
• Normal-isomorph-containing subgroup