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 $\infty$?



The order of $(a_{1},a_{2}, \cdots , a_{n}) \in \oplus_{i} G_{i}$ is the least common multiple of the orders of the $a_{i} \in G_i$, so it is bounded by the product of the orders. Does this help you answer your question?

  • $\begingroup$ So, if one of a_i has order $\infty$, we are done. is that correct? $\endgroup$ – ILoveMath Sep 23 '12 at 21:23
  • $\begingroup$ That is correct. $\endgroup$ – Elchanan Solomon Sep 23 '12 at 21:24

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