Automorph-dominating subgroup

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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:

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

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