Normal subgroup having no nontrivial homomorphism from its quotient group
From Groupprops
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]
Contents
Definition
A subgroup of a group
is termed a normal subgroup having no nontrivial homomorphism from its quotient group if
is a normal subgroup of
and there is no nontrivial homomorphism of groups from the quotient group
to
.
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 |