# Function restriction formalism chart

From Groupprops

Subgroup property | Function restriction expression | Endo-invariance | Balanced | Invariance | Left-inner | Quotient-hereditary invariance |
---|---|---|---|---|---|---|

Normal subgroup | Inner Aut, Inner Function | Yes | No | Yes | Yes | Yes |

Characteristic subgroup | Aut Aut, Aut Function | Yes | Yes | Yes | No | No |

Strictly characteristic subgroup | Surj. End Function, Surj. End. End. | Yes | No | Yes | No | Yes |

Fully invariant subgroup | End. End. | Yes | Yes | Yes | No | Yes |

Injective endomorphism-invariant subgroup | Inj. End. End., Inj. end. Inj. End. | Yes | Yes | Yes | No | No |

Central factor | Inner Inner | No | Yes | No | Yes | No |

Transitively normal subgroup | Normal Normal, Inner Normal | No | Yes | No | Yes | No |

Conjugacy-closed normal subgroup | Class Class, Inner Class | No | Yes | No | Yes | ? |

Retraction-invariant subgroup | Retraction Retraction, Retraction Function | Yes | Yes | Yes | No | Yes |

Note:

- Endo-invariance implies join-closed.
`For full proof, refer: Endo-invariance implies join-closed` - Balanced implies transitive. Conversely, any function-restriction-expressible subgroup property that is transitive, must be balanced.
`For full proof, refer: Balanced implies transitive` - Invariance implies intersection-closed.
`For full proof, refer: Invariance implies intersection-closed` - Left-inner implies left-extensibility-stable, that in turn implies intermediate subgroup condition.
`For full proof, refer: Left-extensibility-stable implies intermediate subgroup condition` - Quotient-hereditary invariance implies quotient-transitive