# Locally inner 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 a **locally inner automorphism-balanced subgroup** if the following equivalent conditions are satisfied:

- Every inner automorphism of restricts to a locally inner automorphism of .
- Every locally inner automorphism of restricts to a locally inner automorphism of .

## 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 | is a central factor of if ... | This means that characteristicity is ... | Additional comments |
---|---|---|---|

locally inner automorphism locally inner automorphism | every locally inner automorphism of restricts to a locally inner automorphism of | the balanced subgroup property for locally inner automorphisms | Hence, it is a t.i. subgroup property, both transitive and identity-true |

inner automorphism locally inner automorphism | every inner automorphism of restricts to a locally inner automorphism of |