Normal subgroup having no nontrivial homomorphism from its quotient group

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Definition

A subgroup $H$ of a group $G$ is termed a normal subgroup having no nontrivial homomorphism from its quotient group if $H$ is a normal subgroup of $G$ and there is no nontrivial homomorphism of groups from the quotient group $G/H$ to $H$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal Sylow subgroup	the whole group is a finite group and the subgroup is normal as well as a Sylow subgroup			|FULL LIST, MORE INFO
normal Hall subgroup	the whole group is a finite group and the subgroup is normal as well as a Hall subgroup -- its order and index are relatively prime.			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal subgroup

Related properties

• Normal subgroup having no nontrivial homomorphism to its quotient group

Other incomparable properties

• Characteristic subgroup: See no nontrivial homomorphism from quotient group not implies characteristic