Automorph-dominating subgroup: Difference between revisions

Revision as of 17:35, 21 December 2014

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Definition

A subgroup $H$ of a group $G$ is termed an automorph-dominating subgroup if it satisfies the following equivalent conditions:

For any automorphism $σ$ of $G$ , there exists $g \in G$ such that the automorph $σ (H)$ is contained in the conjugate subgroup $g H g^{- 1}$ .
For any automorphism $σ$ of $G$ , there exists $w \in G$ such that the automorph $σ (H)$ is contained in the conjugate subgroup $w H w^{- 1}$ .

Note that the $w$ for version (2) may not equal the $g$ for version (1).

Note that if $H$ 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
isomorph-dominating subgroup	every isomorphic subgroup is contained in a conjugate subgroup			\|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

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 ==Definition==
-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 it satisfies the following equivalent conditions:
+# 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>.
+# For any [[automorphism]] <math>\sigma</math> of <math>G</math>, there exists <math>w \in G</math> such that the [[automorphic subgroups|automorph]] <math>\sigma(H)</math> is contained in the [[conjugate subgroups|conjugate subgroup]] <math>wHw^{-1}</math>.
+Note that the <math>w</math> for version (2) may not equal the <math>g</math> for version (1).
 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]].