# Fixed-point subgroup of a subgroup of the automorphism group

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 **fixed-point subgroup of a subgroup of the automorphism group** if there is a subgroup of the automorphism group such that is precisely the subset of fixed under the action of .

## Formalisms

This property is the property of being *closed* with respect to a Galois correspondence between the group and its automorphism group, via the fixed point relation.

## Particular cases

Subgroup of automorphism group | Name for corresponding fixed-point subgroup |
---|---|

automorphism group | absolute center |

inner automorphism group | center |

trivial subgroup | whole group |

## Relation with other properties

### Stronger properties

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

c-closed subgroup |

### Weaker properties

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

local powering-invariant subgroup | whenever an element of the subgroup has a unique root in the group, the root is in the subgroup | fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant | ||

powering-invariant subgroup | the subgroup is powered for all primes for which the group is powered |

## Metaproperties

### Intersection-closedness

YES:This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closedABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness