Mike Speciner wrote:
Am I missing something? According to mathworld, a free abelian group is an abelian group with no torsion, seemingly meaning that the only element with finite order is the identity (in this case the zero function). Clearly here, no function other than zero has finite order.
Mathword does badly on this one -- its definition is just fine for finitely generated groups, but not in general. An abelian group G is "free abelian" if there exists a set of generators {g_i, i in I}, for some index set I (not necessarily countable, of course), such that every element can be written as a finite sum of generators and their inverses in exactly one way. The group P in question is precisely a counterexample to the hypothesis "That just means the group is torsion-free, right?" --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.