# Subgroup-cofactorial automorphism-invariant subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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

### For a finite subgroup

Suppose $G$ is a group and $H$ is a finite subgroup. $H$ is termed a subgroup-cofactorial automorphism-invariant subgroup of $G$ if $H$ is invariant under every automorphism $\sigma$ of $G$ of finite order for which every prime divisor of the order of $\sigma$ is a prime divisor of the order of $H$.

### For a periodic subgroup

Suppose $G$ is a group and $H$ is a periodic subgroup, i.e., a subgroup in which every element has finite order. $H$ is termed a subgroup-cofactorial automorphism-invariant subgroup of $G$ if $H$ is invariant under every automorphism $\sigma$ of $G$ of finite order for which every prime divisor of the order of $\sigma$ is a prime divisor of the order of some element of $H$.

### For a subgroup that is not periodic

If $H$ is a subgroup of $G$ with an element of infinite order, we declare $H$ to be subgroup-cofactorial automorphism-invariant if and only if $H$ is a characteristic subgroup of $G$.

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.
ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitive
ABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

### 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

### Intermediate subgroup condition

NO: This subgroup property does not satisfy the intermediate subgroup condition: it is possible to have a subgroup satisfying the property in the whole group but not satisfying the property in some intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition