# Normal subgroup having no nontrivial homomorphism from its quotient group

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## 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