"The product of infinitely many torsion groups will no longer be a torsion group unless all factors have a common finite exponent (which is not the case if we take Prüfer groups)."
How do I interpret the words factors and having a common finite exponent in this sentence?
I'm not interpreting it correctly because I'm thinking of any element $x$ of a Prufer group as a factor and they all have having a common finite exponent $e$ by virtue of the group being torsion.
My only hypotheses are a) I'm misunderstanding the words factor and/or exponent, or b) that the comment relates to the elements and factors of the infinite product/sum rather than of the Prufer group itself?
It means that if $(G_j\,:\, j\in J)$ is a family of groups, then $\prod\limits_{j\in J} G_j$ is a torsion group if and only if $$\exists m>0,\ \forall j\in J,\ \forall x\in G_j,\ x^m=e_j$$
whereas a family of torsion groups generally only satisfies $$\forall j\in J,\ \exists m_j>0,\ \forall x\in G_j,\ x^{m_j}=e_j$$