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Let $G = \prod_\alpha G_\alpha$, where $G_\alpha$ are torsion subgroups. It is obvious that if all but finitely many of the $G_\alpha$ have a uniform bound, then $G$ is torsion. For the other direction, it suffices to show that the product of a countable collection $\{G_i\}_{i\in \mathbb N}$ without a uniform bound is not a torsion group. Let $\{n_i\}_{i\in \mathbb N}$ be an unbounded sequence of natural numbers such that $G_i^{n_i} \neq 1$ for all $i$. For each $i\in \mathbb N$, choose $g_i \in G_i$ such that $g_i^{n_i} \neq 1$. Then the element $$ (g_1, g_2, \dots) \in \prod_{i \in \mathbb N} G_i $$ is not torsion. (责任编辑：) |