# Pure subgroup of torsion-free abelian group not implies direct factor

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., pure subgroup of torsion-free abelian group) neednotsatisfy the second subgroup property (i.e., direct factor)

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## Contents

## Statement

### Statement for pure subgroups

It is possible to have the following:

- is a torsion-free abelian group
- is a pure subgroup of (this means that for any and such that has a solution for , there is a solution for ).
- is
*not*a direct factor of .

### Statement for local powering-invariant subgroups

It is possible to have the following:

- is a torsion-free abelian group
- is a local powering-invariant subgroup of (this means that for any and such that has a
*unique*solution for , that unique solution is in ). - is
*not*a direct factor of .

## Related facts

## Proof

Example where , for some prime number :

Define as the subgroup of with the following countable generating set:

Explicitly, the generators are:

We note that this generating set has the property that, for all :

Say and take as the subgroup .

We therefore get, for all :

### is a torsion-free abelian group

This is obvious from it being a subgroup of .

### The element is not in

If , it must be in the subgroup generated by finitely many generators:

As shown above, therefore, it must be in the subgroup:

The image modulo is a cyclic group of order . It's easy to see that there is no element in this image that equals the image of ; in fact, forcing the first coordinate to map to zero in the quotient also forces the second coordinate to map to zero.

### is a pure subgroup

For this, it suffices to show that any element of the form that is in is also in . First, if is reduced, must be a power of , say , and relatively prime to . Next, the existence of would force the existence of . If , this forces , which we saw is false from above. Thus, , so , and therefore the element .

### There is no complement to in , so is not a direct factor

The quotient group is isomorphic to the first coordinate projection, which is , but there is no element in that has arbitrarily large power roots.