# Quotient-powering-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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This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: powering-invariant subgroup and normal subgroup satisfying the subgroup-to-quotient powering-invariance implication

View other subgroup property conjunctions | view all subgroup properties

## Definition

A normal subgroup of a group is termed a **quotient-powering-invariant subgroup** if, for any prime number such that is a powered for , the quotient group is also powered for .

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

quotient-transitive subgroup property | Yes | quotient-powering-invariance is quotient-transitive | If are such that is quotient-powering-invariant in and is quotient-powering-invariant in , then is quotient-powering-invariant in . |

union-closed subgroup property | Yes | quotient-powering-invariance is union-closed | If are all quotient-powering-invariant subgroups of a group , and their set-theoretic union is a subgroup , then is also a quotient-powering-invariant subgroup of . |

## Relation with other properties

### Stronger properties

### Weaker properties

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

powering-invariant normal subgroup | quotient-powering-invariant implies powering-invariant | powering-invariant and normal not implies quotient-powering-invariant | |FULL LIST, MORE INFO | |

powering-invariant subgroup | quotient-powering-invariant implies powering-invariant | (via powering-invariant normal) | Powering-invariant normal subgroup|FULL LIST, MORE INFO | |

normal subgroup satisfying the subgroup-to-quotient powering-invariance implication | |FULL LIST, MORE INFO | |||

normal subgroup | (by definition) | (via powering-invariant normal) | Powering-invariant normal subgroup|FULL LIST, MORE INFO |

### Properties whose conjunction with powering-invariance implies quotient-powering-invariance

The relevant subgroup property is normal subgroup satisfying the subgroup-to-quotient powering-invariance implication. In fact, the conjunction of this with powering-invariant subgroup precisely gives quotient-powering-invariant subgroup.

This property is implied both by being a central subgroup and by being a normal subgroup contained in the hypercenter.