# Automorphically endomorph-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 automorphically endomorph-dominating if, for any endomorphism $\sigma$ of $G$, there exists an automorphism $\varphi$ of $G$ such that $\sigma(H) \le \varphi(H)$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
endomorph-dominating subgroup image under any endomorphism is contained in one of its conjugate subgroups follows from the fact that conjugate subgroups are automorphic subgroups |FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subgroup whose characteristic closure is fully invariant its characteristic closure is a fully invariant subgroup |FULL LIST, MORE INFO
subgroup whose characteristic core is fully invariant its characteristic core is a fully invariant subgroup |FULL LIST, MORE INFO