Every group is a subgroup of a divisible 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 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

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

  1. 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:

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