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

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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

Statement

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

  • There exists a group K containing G such that K is a \pi-powered nilpotent group and such that for any element x \in K, there exists a \pi-number n (i.e., a number n all whose prime divisors are in \pi) with x^n \in G.
  • Further, if L is any \pi-powered nilpotent group containing G, there is a subgroup K_1 of L containing G such that the embeddings of G in K and K_1 are equivalent. Moreover, this K_1 is precisely the set of x \in L such that there exists a \pi-number n (i.e., a number n all whose prime divisors are in \pi) with x^n \in G.

Related facts

The localization functor

2 X 2 matrix of related ideas

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

References

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