# Every pi-torsion-free nilpotent group can be embedded in a unique minimal pi-powered nilpotent group

From Groupprops

This article gives the statement, and possibly proof, of an embeddability theorem: a result that states that any group of a certain kind can be embedded in a group of a more restricted kind.

View a complete list of embeddability theorems

## Contents

## Statement

Suppose is a set of prime numbers and is a -torsion-free nilpotent group, i.e., is a nilpotent group and it is -torsion-free -- it has no elements of order for any . Then, the following are true:

- There exists a group containing such that is a -powered nilpotent group and such that for any element , there exists a -number (i.e., a number all whose prime divisors are in ) with .
- Further, if is any -powered nilpotent group containing , there is a subgroup of containing such that the embeddings of in and are equivalent. Moreover, this is precisely the set of such that there exists a -number (i.e., a number all whose prime divisors are in ) with .

## Related facts

### The localization functor

The result can be understood in a 2 X 2 matrix of results:

Group assumption on both the starting group and the big group | Divisibility/powering/torsion assumption on the starting group | Divisibility/powering/torsion assumption on the big group | Is this always possible? | Proof |
---|---|---|---|---|

nilpotent group | none | divisible group | No | nilpotent group need not be embeddable in a divisible nilpotent group |

nilpotent group | -torsion-free group (equivalently, -powering-injective group) | -powered group | Yes | every pi-torsion-free nilpotent group can be embedded in a unique minimal pi-powered nilpotent group |

arbitrary group | none | divisible group | Yes | every group is a subgroup of a divisible group, every pi-group is a subgroup of a divisible pi-group |

arbitrary group | -torsion-free group (equivalently, -powering-injective group) | -powered group | No | Powering-injective group need not be embeddable in a rationally powered group |

## References

(Add reference to Khukhro's p-automorphisms book)