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

Related properties

Other incomparable properties