# Powering-invariant subgroup

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]

## Contents

## Definition

A subgroup of a group is termed **powering-invariant** if it satisfies the following equivalent conditions:

No. | Shorthand | A subgroup of a group is powering-invariant in if ... |
---|---|---|

1 | powered over primes the group is powered over | for every prime such that is powered over (i.e., it is uniquely -divisible), is also powered over . |

2 | invariant under well-defined root maps | for every prime such that for all , there exists unique such that , if we denote , we have that the subgroup is invariant under the function . |

3 | invariant under well-defined root maps | for every natural number such that for all , there exists unique such that , if we denote , we have that the subgroup is invariant under the function . |

4 | invariant under well-defined rational power maps | for every rational number (in reduced form) such that for all , there is a unique satisfying if we denote , we have that the subgroup is invariant under the function . |

5 | invariant under well-defined rational power maps (slight variant) | for every rational number (in reduced form) such that every element has a unique root, if we denote by the unique root of , we have that the subgroup is invariant under the function . |

## Metaproperties

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

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

strongly intersection-closed subgroup property | Yes | powering-invariance is strongly intersection-closed | If is a (finite or infinite, possibly empty) collection of powering-invariant subgroups of a group , then the intersection is also powering-invariant. |

intermediate subgroup condition | No | powering-invariance does not satisfy intermediate subgroup condition | It is possible to have groups such that is powering-invariant in but not in . |

finite-join-closed subgroup property | No | powering-invariance is not finite-join-closed (note, however, that powering-invariance is strongly join-closed in nilpotent group) | It is possible to have a group and powering-invariant subgroups and of such that is not powering-invariant in . |

quotient-transitive subgroup property | No | powering-invariance is not quotient-transitive, but see powering-invariant over quotient-powering-invariant implies powering-invariant | It is possible to have groups such that is powering-invariant and normal in and is powering-invariant in but is not powering-invariant in . |

centralizer-closed subgroup property | Yes | powering-invariance is centralizer-closed (follows from c-closed implies powering-invariant) | If is a group and is a powering-invariant subgroup of , then the centralizer is also powering-invariant. In fact, is powering-invariant even if is not. |

commutator-closed subgroup property | No | powering-invariance is not commutator-closed (note, however, that powering-invariance is commutator-closed in nilpotent group) | It is possible to have a group and powering-invariant subgroups of such that the commutator is not powering-invariant. |

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

lower central series condition | No | powering-invariance does not satisfy lower central series condition | It is possible to choose a group , a powering-invariant subgroup of , and a positive integer such that is not a powering-invariant subgroup of . |

## Relation with other properties

### Dual property

For more on the background, see subgroup-quotient duality for groups.

The dual property to this is quotient-powering-invariant subgroup. A subgroup of a group is termed quotient-powering-invariant in if is a normal subgroup of and for every prime number such that is -powered, is also -powered.

The correspondence we use is subgroup quotient group.

### Stronger properties

### Incomparable properties

Property | Meaning | Proof that it does not imply being powering-invariant | Properties for the ambient group for which it does imply being powering-invariant | Proof that being powering-invariant does not imply the property | Property conjunction |
---|---|---|---|---|---|

normal subgroup | invariant under all inner automorphisms | see examples for characteristic or central subgroup below. | finite group or periodic group | any finite non-normal subgroup | powering-invariant normal subgroup |

characteristic subgroup | invariant under all automorphisms | characteristic not implies powering-invariant | abelian group (plus above cases for normal subgroup) and perhaps nilpotent group | any finite non-characteristic subgroup | powering-invariant characteristic subgroup |

central subgroup | contained in the center | follows from subgroup of abelian group not implies powering-invariant | (all cases for normal subgroup) | any non-abelian group as a subgroup of itself | powering-invariant central subgroup |

## Satisfaction by subgroup-defining functions

## 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

A function restriction expression for the property of being a powering-invariant subgroup is as follows:

Rational power map Function

Here, "rational power map" refers to a map of the form where is a rational number for which the map is well-defined (see definitions no. (4) or (5)).

This shows that the property is an invariance property, which immediately implies that it is, among other things, strongly intersection-closed.

The expression can be right-tightened to give a balanced subgroup property:

Rational power map Rational power map

It being a balanced subgroup property immediately shows that it's a transitive subgroup property (since balanced implies transitive).

Above, we could replace "rational power map" throughout to a subset of rational power maps, such as prime root maps or integer root maps (see definitions (2) or (3)). We chose "rational power map" because the well-defined rational power maps on a group form a subquotient of the multiplicative rationals and are the largest such permitted subquotient.