Completely distinguished subgroup: Difference between revisions
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===Weaker properties=== | ===Weaker properties=== | ||
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! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
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| [[Stronger than::surjective endomorphism-balanced subgroup]] || || || || {{intermediate notions short|surjective endomorphism-balanced subgroup|completely distinguished subgroup}} | |||
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| [[Stronger than::strictly characteristic subgroup]] || || || || {{intermdiate notions shirt|strictly characteristic subgroup|completely distinguished subgroup}} | |||
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| [[Stronger than::characteristic subgroup]] || || || || {{intermediate notions short|characteristic subgroup|completely distinguished subgroup}} | |||
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==Metaproperties== | ==Metaproperties== | ||
Latest revision as of 04:07, 3 April 2013
This article defines a subgroup property related to (or which arises in the context of): combinatorial group theory
View other subgroup properties related to combinatorial group theory|View other terms related to combinatorial group theory | View all subgroup properties
Definition
Symbol-free definition
A subgroup of a group is termed completely distinguished if for every surjective endomorphism from the group to itself, the subgroup equals its complete pre-image.
Definition with symbols
A subgroup of a group is termed completely distinguished in if it satisfies the following equivalent conditions:
- For any surjective endomorphism ,
- is a distinguished subgroup (also termed a strictly characteristic subgroup) and is a Hopfian group
Relation with other properties
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| surjective endomorphism-balanced subgroup | |FULL LIST, MORE INFO | |||
| strictly characteristic subgroup | Template:Intermdiate notions shirt | |||
| characteristic subgroup | |FULL LIST, MORE INFO |
Metaproperties
Transitivity
This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this 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 transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity
Intersection-closedness
YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness