Automorph-dominating subgroup: Difference between revisions
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A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''automorph-dominating subgroup''' if, for any [[automorphism]] <math>\sigma</math> of <math>G</math>, there exists <math>g \in G</math> such that the [[automorphic subgroups|automorph]] <math>\sigma(H)</math> is contained in the [[conjugate subgroups|conjugate subgroup]] <math>gHg^{-1}</math>. | A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed an '''automorph-dominating subgroup''' if, for any [[automorphism]] <math>\sigma</math> of <math>G</math>, there exists <math>g \in G</math> such that the [[automorphic subgroups|automorph]] <math>\sigma(H)</math> is contained in the [[conjugate subgroups|conjugate subgroup]] <math>gHg^{-1}</math>. | ||
Note that if <math>H</math> is a [[co-Hopfian group]] (i.e. it does not contain any proper subgroup isomorphic to it) this property is equivalent to being an [[automorph-conjugate subgroup]]. | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 17:30, 21 December 2014
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]
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Definition
A subgroup of a group is termed an automorph-dominating subgroup if, for any automorphism of , there exists such that the automorph is contained in the conjugate subgroup .
Note that if is a co-Hopfian group (i.e. it does not contain any proper subgroup isomorphic to it) this property is equivalent to being an automorph-conjugate subgroup.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| endomorph-dominating subgroup | every image under an endomorphism is contained in a conjugate | (obvious) | follows from characteristic not implies fully invariant -- any characteristic subgroup that is not fully invariant will do | |FULL LIST, MORE INFO |
| homomorph-dominating subgroup | every image under a homomorphism is contained in a conjugate | (via endomorph-dominating) | (via endomorph-dominating) | |FULL LIST, MORE INFO |
| automorph-conjugate subgroup | every automorphic subgroup equals a conjugate | (obvious) | follows from endomorph-dominating not implies automorph-conjugate | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| closure-characteristic subgroup | normal closure is a characteristic subgroup | |FULL LIST, MORE INFO | ||
| core-characteristic subgroup | normal core is a characteristic subgroup | |FULL LIST, MORE INFO |