Locally inner automorphism-balanced subgroup

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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]

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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			|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

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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 $H$