# Locally inner automorphism-balanced subgroup

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## Contents

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

A subgroup $H$ of a group $G$ is termed a locally inner automorphism-balanced subgroup if the following equivalent conditions are satisfied:

1. Every inner automorphism of $G$ restricts to a locally inner automorphism of $H$.
2. Every locally inner automorphism of $G$ restricts to a locally inner automorphism of $H$.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
central factor every inner automorphism of the whole group restricts to an inner automorphism (obvious) locally inner automorphism-balanced not implies central factor |FULL LIST, MORE INFO
direct factor factor in an internal direct product Central factor|FULL LIST, MORE INFO

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
transitively normal subgroup every normal subgroup of it is normal in the whole group |FULL LIST, MORE INFO
normal subgroup every inner automorphism of the whole group sends the subgroup to itself |FULL LIST, MORE INFO

## Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

### Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property
Function restriction expression $H$ is a central factor of $G$ if ... This means that characteristicity is ... Additional comments
locally inner automorphism $\to$ locally inner automorphism every locally inner automorphism of $G$ restricts to a locally inner automorphism of $H$ the balanced subgroup property for locally inner automorphisms Hence, it is a t.i. subgroup property, both transitive and identity-true
inner automorphism $\to$ locally inner automorphism every inner automorphism of $G$ restricts to a locally inner automorphism of $H$