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I will talk about a mysterious little exercise that I have been struggling with for the last couple of weeks and so far have been unable to solve.

If A and B are abelian groups, A a pure subgroup of B, and A' is a subgroup of A of index 2, then A' is contained in a subgroup B' of B of index 2. This follows easily from the fact that Z/2 is pure-injective. (The same holds for index p, p a prime, and for many more general cases.) On the other hand, it seems that such an easy statement that it should be provable just from the definition of purity.

Purity for abelian groups is easy (but the equivalence with the usual definition requires AC): A a pure subgroup of B if A is relatively divisible in B; that is, if for a in A and natural number n greater than 1 there is b in B such that nb=a, then in fact we can find such b in A.

There are basically two equivalent families of definitions of pure-injectivity, one in terms of some form of equational compactness, and the other in some categorical form, including the main one of injectivity over pure embeddings. Again, the complete chain of equivalences involves the use of the Axiom of Choice: the fact that Z/2 is pure-injective is not completely elementary!

I will explain all this in detail and show various unsuccessful proofs of the desired result---or maybe with luck, a successful proof!