# Automorphically endomorph-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 **automorphically endomorph-dominating** if, for any endomorphism of , there exists an automorphism of such that .

## 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 |