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

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 $H$ of a group $G$ is termed a fixed-point subgroup of a subgroup of the automorphism group if there is a subgroup $B$ of the automorphism group $\operatorname{Aut}(G)$ such that $H$ is precisely the subset of $G$ fixed under the action of $B$.

## 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 $n^{th}$ 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-closed
ABOUT 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