Consider two groups $G$ and $H$. Let $G\times H$ be their direct product. Let $(g_1,h_1),(g_2,h_2)\in G\times H$.

What does it look like when $G\times H$ is abelian?

Does that mean

  1. $(g_1g_2,h_1h_2)=(g_2g_1, h_2h_1)$ or

  2. $(g_1g_2, h_1h_2)=(h_1h_2, g_1g_2)$?

  • 3
    $\begingroup$ It's option 1. Option 2 doesn't make sense. You're equating apples to oranges. $\endgroup$ – Evan Aad Oct 23 '16 at 20:29
  • $\begingroup$ 1) is the right case. $\endgroup$ – Bernard Oct 23 '16 at 20:29

It is the first. In your second option, $(h_1h_2,g_1g_2) \notin G \times H$.


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