# Automorph-dominating subgroup

From Groupprops

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]

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

## Definition

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

- For any automorphism of , there exists such that the automorph is contained in the conjugate subgroup .
- For any automorphism of , there exists such that the automorph contains the conjugate subgroup .

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

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) | 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 |