Endomorphism image: Difference between revisions
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| [[Stronger than::divisibility-closed subgroup]] || if every element of the group has a <math>n^{th}</math> root, so does every element of the subgroup. || [[endomorphism image implies divisibility-closed]] || any subgroup of a finite group that is not an endomorphism image works. || {{intermediate notions short|divisibility-closed subgroup|endomorphism image}} | | [[Stronger than::divisibility-closed subgroup]] || if every element of the group has a <math>n^{th}</math> root, so does every element of the subgroup. || [[endomorphism image implies divisibility-closed]] || any subgroup of a finite group that is not an endomorphism image works. || {{intermediate notions short|divisibility-closed subgroup|endomorphism image}} | ||
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| [[Stronger than::powering-invariant subgroup]] || if every element of the group has a unique <math>n^{th}</math> root, so does every element of the subgroup. || || || {{intermediate notions short|powering-invariant subgroup|endomorphism image}} | | [[Stronger than::powering-invariant subgroup]] || if every element of the group has a unique <math>n^{th}</math> root, so does every element of the subgroup. || (via divisibility-closed) || (via divisibility-closed) || {{intermediate notions short|powering-invariant subgroup|endomorphism image}} | ||
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Latest revision as of 18:27, 15 February 2013
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 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 | |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 root, so does every element of the subgroup. | 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 root, so does every element of the subgroup. | (via divisibility-closed) | (via divisibility-closed) | |FULL LIST, MORE INFO |
Related properties
- Endomorphism kernel: Kernel of an endomorphism.