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 is a group. Then, there exists a group containing as a subgroup such that is a divisible group, i.e., is divisible for all primes.
Related facts
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 |
Facts used
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 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 .
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)