# Automorphism-faithful subgroup

From Groupprops

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

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]

## Definition

### Symbol-free definition

A subgroup of a group is termed **automorphism-faithful** if any nontrivial automorphism of the whole group that restricts to an automorphism of the subgroup, in fact restricts to a *nontriival* automorphism of the subgroup.

### Definition with symbols

A subgroup of a group is termed **automorphism-faithful** in if for any nontrivial automorphism of such that , the restriction of to is a nontrivial automorphism of .

## Relation with other properties

### Stronger properties

- Centralizer-free normal subgroup:
`For full proof, refer: Centralizer-free and normal implies automorphism-faithful` - Centralizer-free characteristic subgroup
- Automorphism-faithful normal subgroup
- Automorphism-faithful characteristic subgroup