Every pi-torsion-free nilpotent group can be embedded in a unique minimal pi-powered nilpotent group
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
- 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 .
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|
(Add reference to Khukhro's p-automorphisms book)