Normal subgroup having no nontrivial homomorphism to 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 a normal subgroup having no nontrivial homomorphism to its quotient group if
is a normal subgroup of
and there is no nontrivial homomorphism from
to its quotient group
.
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
transitive subgroup property | No | no nontrivial homomorphism to quotient group is not transitive | It is possible to have groups ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
trim subgroup property | Yes | (obvious reasons) | In any group, the trivial subgroup and the whole group satisfy the condition of being normal subgroups with no nontrivial homomorphism to their respective quotient groups. |
intermediate subgroup condition | Yes | no nontrivial homomorphism to quotient group satisfies intermediate subgroup condition | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness | Intermediate notions |
---|---|---|---|---|
normal Sylow subgroup | the whole group is a finite group and the subgroup is both normal and a Sylow subgroup. | |FULL LIST, MORE INFO | ||
normal Hall subgroup | the whole group is a finite group and the subgroup is both normal and a Hall subgroup -- its order and index are relatively prime. | |FULL LIST, MORE INFO | ||
fully invariant direct factor | both a fully invariant subgroup and a direct factor of the whole group. | |FULL LIST, MORE INFO |
Weaker properties
Related properties
- Normal subgroup having no nontrivial homomorphism from its quotient group
- Normal subgroup having no common composition factor with its quotient group
Other incomparable properties
- Complemented normal subgroup: For full proof, refer: No nontrivial homomorphism to quotient group not implies complemented normal
Metaproperties
Transitivity
NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
ABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity
For full proof, refer: No nontrivial homomorphism to quotient group is not transitive
Trimness
This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties
Intermediate subgroup condition
YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition
If , with
normal in
and no nontrivial homomorphism from
to
,
is also normal in
with no nontrivial homomorphism from
to
.
For full proof, refer: No nontrivial homomorphism to quotient group satisfies intermediate subgroup condition