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

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

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)