# Local powering-invariant subgroup

From Groupprops

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]

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

## Definition

A subgroup of a group is termed a **local powering-invariant subgroup** if the following hold:

- Whenever and are such that there is a unique such that , we must have .
- Whenever and is a prime number such that there is a unique such that , we must have .

### Equivalence of definitions

`Further information: equivalence of definitions of local powering-invariant subgroup`

## Metaproperties

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

transitive subgroup property | Yes | local powering-invariance is transitive | Suppose are groups such that is local powering-invariant in and is local powering-invariant in . Then is local powering-invariant in . |

strongly intersection-closed subgroup property | Yes | local powering-invariance is strongly intersection-closed | Suppose are all local powering-invariant subgroups of a group . Then, the intersection of subgroups is also a local powering-invariant subgroup of . |

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

finite-join-closed subgroup property | No | local powering-invariance is not finite-join-closed | It is possible to have a group and local powering-invariant subgroups and of such that the join of subgroups is not local powering-invariant. |

centralizer-closed subgroup property | Yes | follows from c-closed implies local powering-invariant | Suppose is a group and is a local powering-invariant subgroup of . Then, the centralizer is also a local powering-invariant subgroup of . In fact, we do not even need to be local powering-invariant -- the assumption is redundant. |

commutator-closed subgroup property | No | local powering-invariance is not commutator-closed | It is possible to have a group and subgroups of such that both and are local powering-invariant in but the commutator is not. |

image condition | No | local powering-invariance does not satisfy image condition | It is possible to have a group , a local powering-invariant subgroup of , and a surjective homomorphism such that is not local powering-invariant in . |

## Relation with other properties

### Stronger properties

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

finite subgroup | finite implies local powering-invariant | |||

periodic subgroup | ||||

fixed-point subgroup of a subgroup of the automorphism group | fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant | |||

c-closed subgroup | (via fixed-point subgroup of a subgroup of the automorphism group) | |||

local divisibility-closed subgroup | |FULL LIST, MORE INFO | |||

retract | has a normal complement | (via local divisibility-closed) | (via local divisibility-closed) | Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Verbally closed subgroup|FULL LIST, MORE INFO |

direct factor | normal subgroup with normal complement | (via retract) | (via retract) | Intermediately local powering-invariant subgroup, Local divisibility-closed subgroup, Retract, Verbally closed subgroup|FULL LIST, MORE INFO |

### Weaker properties

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

powering-invariant subgroup |

### Incomparable properties

Property | Meaning | Proof that it does not imply being local powering-invariant | Proof that being local powering-invariant does not imply it | Properties stronger than both | Properties weaker than both |
---|---|---|---|---|---|

subgroup of finite index | has finite index in the whole group | finite index not implies local powering-invariant | any direct factor where the other direct factor is infinite. | |FULL LIST, MORE INFO | Powering-invariant subgroup|FULL LIST, MORE INFO |

complemented normal subgroup | normal subgroup with a permutable complement | complemented normal not implies local powering-invariant | any non-normal subgroup of a finite group will do. | Direct factor|FULL LIST, MORE INFO | Powering-invariant subgroup|FULL LIST, MORE INFO |