Subgroup-cofactorial automorphism-invariant subgroup

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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.

Relation with other properties

Stronger properties

Weaker properties

Incomparable properties

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

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