# Normal-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]

## Definition

A subgroup of a group is termed a **normal-potentially fully invariant subgroup** of if there exists a group containing as a normal subgroup, such that is a fully invariant subgroup of .

## Formalisms

### In terms of the normal-potentially operator

This property is obtained by applying the normal-potentially operator to the property: fully invariant subgroup

View other properties obtained by applying the normal-potentially operator

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Fully invariant subgroup | |FULL LIST, MORE INFO | |||

Central subgroup of finite group | Normal-potentially verbal subgroup|FULL LIST, MORE INFO | |||

Normal-potentially verbal subgroup | |FULL LIST, MORE INFO | |||

Fully invariant-potentially fully invariant subgroup | |FULL LIST, MORE INFO |

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Potentially fully invariant subgroup | |FULL LIST, MORE INFO | |||

Normal-potentially characteristic subgroup | |FULL LIST, MORE INFO | |||

Normal subgroup | Potentially fully invariant subgroup|FULL LIST, MORE INFO |