Every group is a subgroup of a divisible 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 $G$ is a group. Then, there exists a group $K$ containing $G$ as a subgroup such that $K$ is a divisible group, i.e., $K$ is divisible for all primes.

Related facts

Every pi-group is a subgroup of a divisible pi-group

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

Facts used

Diagonal subgroup of a wreath product with a cyclic permutation group is divisible by all primes dividing the order of the cyclic group

Proof

Note that although the proof below uses primorials, we can write a similar proof that uses any sequence with the property that every prime number occurs as a divisor of infinitely many numbers in the sequence. Thus, the sequence of factorials could be used instead.

Consider the sequence of primorials: a sequence $(a_{n})_{n\in \mathbb {N} }$ where $a_{n}$ is the product of the first $n$ primes. The sequence proceeds as follows:

$2,6,30,210,\dots ,$

Now, define a sequence of groups with homomorphisms as follows:

$G_{0}$ is the original group.
For each nonnegative integer $i$ , $G_{i+1}$ is defined as the external wreath product of the cyclic group $G_{i}$ with the cyclic group of order $a_{i+1}$ acting as a cyclic permutation group. The homomorphism from $G_{i}$ to $G_{i+1}$ embeds $G_{i}$ as the diagonal subgroup of a wreath product.

The direct limit of the sequence of injective homomorphisms:

$G_{0}\to G_{1}\to G_{2}\to \dots$

is the desired divisible group containing $G$ .

References

Adjunction of elements to groups by B. H. Neumann, Journal of the London Mathematical Society, ISSN 14697750 (online), ISSN 00246107 (print), Volume 18, Page 12 - 20(Year 1943): ^{Official copy}^{More info}
Wreath products and p-groups by Gilbert Baumslag, Proceedings of the Cambridge Philosophical Society, Volume 55, Page 224 - 231(Year 1959): ^{ungated PDF}^{More info}, Corollary 4.3 (Page 14 of the paper)