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
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|
- Diagonal subgroup of a wreath product with a cyclic permutation group is divisible by all primes dividing the order of the cyclic group
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 where is the product of the first primes. The sequence proceeds as follows:
Now, define a sequence of groups with homomorphisms as follows:
- is the original group.
- For each nonnegative integer , is defined as the external wreath product of the cyclic group with the cyclic group of order acting as a cyclic permutation group. The homomorphism from to embeds as the diagonal subgroup of a wreath product.
The direct limit of the sequence of injective homomorphisms:
is the desired divisible group containing .
- 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 copyMore info
- Wreath products and p-groups by Gilbert Baumslag, Proceedings of the Cambridge Philosophical Society, Volume 55, Page 224 - 231(Year 1959): ungated PDFMore info, Corollary 4.3 (Page 14 of the paper)