Locally inner automorphism

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 Locally inner automorphism, all facts related to Locally inner automorphism) |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 an automorphism property, viz a property of group automorphisms. Hence, it also defines a function property (property of functions from a group to itself)
View other automorphism properties OR View other function properties

Definition

QUICK PHRASES: inner on finite tuples, inner on finitely generated subgroups

Definition with symbols

An automorphism $\sigma$ of a group $G$ is termed a locally inner automorphism if, for any elements $x_{1},x_{2},\dots ,x_{n}\in G$, there exists $g\in G$ such that $gx_{i}g^{-1}=\sigma (x_{i})$ for $1\leq i\leq n$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Inner automorphism

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Class-preserving automorphism	sends every element to within its conjugacy class	locally inner implies class-preserving	class-preserving not implies locally inner	|FULL LIST, MORE INFO
Center-fixing automorphism	fixes every element of the center	(via class-preserving)	(via class-preserving)	|FULL LIST, MORE INFO
IA-automorphism	induces identity map on the abelianization	(via class-preserving)	(via class-preserving)	|FULL LIST, MORE INFO
Normal automorphism	sends every normal subgroup to itself			|FULL LIST, MORE INFO
Weakly normal automorphism	sends every normal subgroup to a subgroup of itself			|FULL LIST, MORE INFO