Endomorphism image
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 an endomorphism image if there exists an endomorphism of
whose image is precisely the subgroup
.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
direct factor | Retract|FULL LIST, MORE INFO | |||
retract | image of an idempotent endomorphism | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
divisibility-closed subgroup | if every element of the group has a ![]() |
endomorphism image implies divisibility-closed | any subgroup of a finite group that is not an endomorphism image works. | |FULL LIST, MORE INFO |
powering-invariant subgroup | if every element of the group has a unique ![]() |
(via divisibility-closed) | (via divisibility-closed) | |FULL LIST, MORE INFO |
Related properties
- Endomorphism kernel: Kernel of an endomorphism.