Automorphism-faithful subgroup

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

For its use outside the wiki, please define the term when using it. If you are aware of an equivalent standard term, please leave a comment on the talk page
VIEW: Definitions built on this | Facts about this: (facts closely related to Automorphism-faithful subgroup, all facts related to Automorphism-faithful subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
Learn more about terminology local to the wiki | view a complete list of such terminology

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]

|Get subgroup property lookup help |Get exploration suggestions
VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

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 $H$.

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

Weaker properties

• Coprime automorphism-faithful subgroup