I'm supposed to find a non-trivial group G such that G is isomorphic to G x G.(adsbygoogle = window.adsbygoogle || []).push({});

I know G must be infinite, since if G had order n, then G x G would have order n^2. So, after some thought, I came up with the following. Z is isomorphic to Z x Z.

My reasoning is similar to the oft-seen proof that the rationals are countable.

Picture a grid with dots representing each element in Z x Z. Now, starting at the origin, trace a circuitous path (in any direction, but always a tight spiral) and define a map that sends 0 to (0,0), 1 to the next point, -1 to the next point, 2 to the next point, etc.

Is it enough to describe this map in the way I have, or do I need further information (or am I wrong?)

Thanks.

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# Abstract Algebra - Direct Product Question

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