# Automorphism-faithful subgroup

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 $H$ of a group $G$ is termed automorphism-faithful in $G$ if for any nontrivial automorphism $\sigma$ of $G$ such that $\sigma(H) = H$, the restriction of $\sigma$ to $H$ is a nontrivial automorphism of $H$.