# IA-automorphism-balanced subgroup

From Groupprops

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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]

## 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

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 | is an IA-automorphism-balanced subgroup of if ... | This means that being IA-automorphism-balanced is ... | Additional comments |
---|---|---|---|

IA-automorphism IA-automorphism | every IA-automorphism of restricts to a IA-automorphism of | 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

## 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 transitiveABOUT 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