# Completely divisibility-closed 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

Suppose is a group. A subgroup of is termed **completely divisibility-closed** if the following holds: for any prime number such that is -divisible, and any , *all* roots of in lie inside .

## Metaproperties

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

transitive subgroup property | Yes | complete divisibility-closedness is transitive | If are groups such that is completely divisibility-closed in and is completely divisibility-closed in , then is completely divisibility-closed in . |

strongly intersection-closed subgroup property | Yes | complete divisibility-closedness is strongly intersection-closed | If are completely divisibility-closed subgroups of , so is . |

## Relation with other properties

### Stronger properties

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

completely divisibility-closed normal subgroup | completely divisibility-closed and normal; equivalently, the quotient is torsion-free for any prime for which the whole group is divisible. | |FULL LIST, MORE INFO | ||

kernel of a bihomomorphism | kernel of a bihomomorphism implies completely divisibility-closed | |||

intersection of kernels of bihomomorphisms | intersection of kernels of bihomomorphisms implies completely divisibility-closed | |FULL LIST, MORE INFO | ||

kernel of a multihomomorphism | kernel of a multihomomorphism implies completely divisibility-closed | Intersection of kernels of bihomomorphisms|FULL LIST, MORE INFO | ||

subgroup of finite group | ||||

subgroup of periodic group |

### Weaker properties

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

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

powering-invariant subgroup | Divisibility-closed subgroup|FULL LIST, MORE INFO |