# Center-fixing automorphism-invariant subgroup

From Groupprops

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]

## Contents

## Definition

A subgroup of a group is termed a **center-fixing automorphism-invariant subgroup** if every center-fixing automorphism of the whole group sends the subgroup to within itself.

## Metaproperties

Metaproperty | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

strongly intersection-closed subgroup property | Yes | Follows from invariance implies strongly intersection-closed | If are all center-fixing automorphism-invariant subgroups of , then so is the intersection of subgroups . |

strongly join-closed subgroup property | Yes | Follows from endo-invariance implies strongly join-closed | If are all center-fixing automorphism-invariant subgroups of , then so is the join of subgroups . |

trim subgroup property | Yes | direct from definition | In any group , both the whole group and the trivial subgroup are center-fixing automorphism-invariant. |

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

characteristic subgroup | invariant under all automorphisms |
characteristic implies center-fixing automorphism-invariant | center-fixing automorphism-invariant not implies characteristic | |FULL LIST, MORE INFO |

central subgroup | contained in the center | central implies center-fixing automorphism-invariant | center-fixing automorphism-invariant not implies central | |FULL LIST, MORE INFO |

center-fixing automorphism-balanced subgroup | every center-fixing automorphism of the whole group restricts to a center-fixing automorphism of the subgroup. | |FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal subgroup | invariant under all inner automorphisms | |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 center-fixing automorphism-invariant subgroup of if ... | This means that being center-fixing automorphism-invariant is ... | Additional comments |
---|---|---|---|

center-fixing automorphism function | every center-fixing automorphism of sends every element of to within | the invariance property for center-fixing automorphisms | On account of this, it is a strongly intersection-closed subgroup property. |

center-fixing automorphism endomorphism | every center-fixing automorphism of restricts to an endomorphism of | the endo-invariance property for center-fixing automorphisms; i.e., it is the invariance property for center-fixing automorphism, which is a property stronger than the property of being an endomorphism | On account of this, it is a strongly join-closed subgroup property. |

center-fixing automorphism automorphism | every center-fixing automorphism of restricts to an automorphism of | the auto-invariance property for center-fixing automorphisms; i.e., it is the invariance property for center-fixing automorphism, which is a group-closed property of automorphisms |