# Potentially fully 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 fully invariant subgroup|Find other variations of fully invariant subgroup |

## Definition

### Symbol-free definition

A subgroup of a group is termed **potentially fully invariant** if there is an embedding of the bigger group in some group such that, in that embedding the subgroup becomes fully invariant.

### Definition with symbols

A subgroup of a group is termed **potentially fully invariant** in if there exists a group containing such that is fully invariant in .

## Formalisms

### In terms of the potentially operator

This property is obtained by applying the potentially operator to the property: fully characteristic subgroup

View other properties obtained by applying the potentially operator

The property of being potentially fully invariant is obtained by applying the potentially operator to the property of being fully invariant. The potentially operator is an idempotent ascendant monotone operator.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

fully invariant subgroup | invariant under all endomorphisms | (by definition) | (cyclic normal subgroup examples) | |

verbal subgroup | image of a word map | (via fully invariant) | (via fully invariant) | |

potentially verbal subgroup | can be verbal inside a bigger group | follows from verbal implies fully invariant | (unclear) | |

normal-potentially fully invariant subgroup | can be fully invariant in a bigger group in which the original ambient group is normal | (by definition) | (unclear) | |

central subgroup of finite group | central implies potentially fully invariant in finite | any non-abelian group as a subgroup of itself | ||

cyclic normal subgroup of a finite group | (via homocyclic normal) | |||

homocyclic normal subgroup of a finite group | homocyclic normal implies potentially fully invariant in finite | |||

fully normalized potentially fully invariant subgroup | also a fully normalized subgroup |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

normal subgroup | potentially fully invariant implies normal | normal not implies potentially fully invariant |

### Incomparable properties

## Metaproperties

### Transitivity

NO:This subgroup property isnottransitive: a subgroup with this property in a subgroup with this property, need not have the property in the whole groupABOUT THIS PROPERTY: View variations of this property that are transitive|View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of subgroup properties that are not transitive|View facts related to transitivity of subgroup properties | View a survey article on disproving transitivity