Let $(a_1,a_2,\cdots,a_n) \in G_1 \oplus \cdots \oplus G_n$. Give a necessary and sufficient condition for $|(a_1,a_2,\dots,a_n)| = \infty$.
I think if one $a_i$ has order $\infty$, then this would give a necessary condition. It is sufficient for the order of the direct product to be $\infty$ when only one of the $a_i$ has order $\infty$?
Thanks,