# Automorph-dominating subgroup

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]

## Contents

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## 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) Isomorph-dominating subgroup|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