# 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 is a group and is a finite subgroup. is termed a **subgroup-cofactorial automorphism-invariant subgroup** of if is invariant under every automorphism of of finite order for which every prime divisor of the order of is a prime divisor of the order of .

### For a periodic subgroup

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

### For a subgroup that is not periodic

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

## Relation with other properties

### Stronger properties

- Characteristic subgroup
- Cofactorial automorphism-invariant subgroup
- Sub-cofactorial automorphism-invariant subgroup

### Weaker properties

- Left-transitively 2-subnormal subgroup:
`For full proof, refer: Subgroup-cofactorial automorphism-invariant implies left-transitively 2-subnormal` - 2-subnormal subgroup
- Subnormal subgroup

### Incomparable properties

- Normal subgroup:
`For full proof, refer: Normal not implies subgroup-cofactorial automorphism-invariant, Subgroup-cofactorial automorphism-invariant not implies normal`

## 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 transitiveABOUT 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-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

### Intermediate subgroup condition

NO:This subgroup property doesnotsatisfy 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 conditionABOUT INTERMEDIATE SUBGROUP CONDITION: View other subgroup properties not satisfying intermediate subgroup condition| View facts about intermediate subgroup condition