Intermediately fully 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

Symbol-free definition

A subgroup of a group is termed intermediately fully invariant or intermediately fully characteristic if it is fully invariant in every intermediate subgroup of the group containing it.

Definition with symbols

A subgroup $H$ of a group $G$ is termed intermediately fully invariant or intermediately fully characteristic in $G$ if, for any intermediate subgroup $K$ of $G$, $H$ is fully characteristic in $K$: for any endomorphism $\phi$ of $K$, $\phi (H)\le H$.

Relation with other properties

Stronger properties

• Homomorph-containing subgroup
• Subhomomorph-containing subgroup
• Variety-containing subgroup

Weaker properties

• Fully invariant subgroup
• Intermediately characteristic subgroup
• Characteristic subgroup

Metaproperties

Transitivity

NO: This subgroup property is not transitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole group
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An intermediately fully characteristic subgroup of an intermediately fully characteristic subgroup need not be intermediately fully characteristic. For full proof, refer: Intermediate full invariance is not transitive

Join-closedness

YES: This subgroup property is join-closed: an arbitrary (nonempty) join of subgroups with this property, also has this property.
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ABOUT JOIN-CLOSEDNESS: View all join-closed subgroup properties (or, strongly join-closed properties) | View all subgroup properties that are not join-closed | Read a survey article on proving join-closedness | Read a survey article on disproving join-closedness

An arbitrary join of intermediately fully characteristic subgroups is intermediately fully characteristic. This follows from the fact that the intermediately operator preserves the property of being closed under joins. For full proof, refer: Intermediate full invariance is strongly join-closed