Projectivity-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

A subgroup of a group is termed projectivity-invariant if it is invariant under every projectivity of the group, i.e., every projectivity of the group maps the subgroup bijectively to itself.

Formalisms

Function restriction expression

This subgroup property is a function restriction-expressible subgroup property: it can be expressed by means of the function restriction formalism, viz there is a function restriction expression for it.
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Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Lattice automorphism-invariant subgroup	invariant under all lattice automorphisms of the lattice of subgroups			|FULL LIST, MORE INFO
Order-unique subgroup	unique subgroup of its order			|FULL LIST, MORE INFO
Projective-isomorph-free subgroup	no other subgroup such that there is a projectivity between the two subgroups			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Characteristic subgroup	invariant under all automorphisms			|FULL LIST, MORE INFO
Normal subgroup	invariant under all inner automorphisms			|FULL LIST, MORE INFO