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]

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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

• 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.
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Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
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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.
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