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]

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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