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) need not satisfy 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.