IA-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 of a group is termed an IA-automorphism-balanced subgroup or IA-balanced subgroup if every IA-automorphism of the whole group restricts to an IA-automorphism of the subgroup.

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 an IA-automorphism-balanced subgroup of $G$ if ...	This means that being IA-automorphism-balanced is ...	Additional comments
IA-automorphism $\to$ IA-automorphism	every IA-automorphism of $G$ restricts to a IA-automorphism of $H$	the balanced subgroup property for IA-automorphisms	Hence, it is a t.i. subgroup property, both transitive and identity-true

Relation with other properties

Stronger properties

• IA-automorphism-invariant direct factor
• Perfect IA-automorphism-invariant subgroup
• IA-automorphism-invariant subgroup whose commutator subgroup equals its intersection with whole commutator subgroup

Weaker properties

• Normal subgroup
• IA-automorphism-invariant subgroup
• Subgroup with canonical abelianization

Metaproperties

Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

Trimness

This subgroup property is trim -- it is both trivially true (true for the trivial subgroup) and identity-true (true for a group as a subgroup of itself).
View other trim subgroup properties | View other trivially true subgroup properties | View other identity-true subgroup properties