Locally inner automorphism

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BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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_ig^{-1} = \sigma(x_i) for 1 \le i \le 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) Class-preserving automorphism|FULL LIST, MORE INFO
IA-automorphism induces identity map on the abelianization (via class-preserving) (via class-preserving) Automorphism that preserves conjugacy classes for a generating set, Class-preserving automorphism|FULL LIST, MORE INFO
Normal automorphism sends every normal subgroup to itself Class-preserving automorphism|FULL LIST, MORE INFO
Weakly normal automorphism sends every normal subgroup to a subgroup of itself Class-preserving automorphism|FULL LIST, MORE INFO