# Projectivity-invariant subgroup

From Groupprops

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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]

This is a variation of characteristic subgroup|Find other variations of characteristic subgroup | Read a survey article on varying characteristic subgroup

## 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.

Find other function restriction-expressible subgroup properties | View the function restriction formalism chart for a graphic placement of this property

## 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 |