Exercise 21, § 11

Let $G$ and $H$ be two commutative torsion groups such that for every integer $n$, the equation $nx = 0$ has the same finite number of solutions in $G$ and in $H$. Show that $G$ is isomorphic to $H$. (Decompose $G$ and $H$ into direct sums of $p$-torsion groups and apply Ex. 20.) Let $G = \bigoplus_{i \in \mathbf{N}} \mathbf{Z}/p^i\mathbf{Z}$ and $H = \bigoplus_{i \in \mathbf{N}} \mathbf{Z}/p^{2i}\mathbf{Z}$. Show that $G$ and $H$ are not isomorphic and that for each integer $n$ the cardinal of the set of solutions of the equation $nx = 0$ is the same in $G$ and in $H$.