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The fundamental group of the torus is isomorphic to $\mathbb{Z}\oplus\mathbb{Z}$.

I know that the $\oplus$ symbol is the exclusive or symbol but I don't understand how two of the same sets are XOR to each other.

Sorry if this is a very simple question.

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    $\begingroup$ Try to forget "XOR" in this context :) It's just that the symbol was used both for "XOR" and the "direct sum", but the former concept is not being used here. Welcome to the wonderful world of recycled symbols. $\endgroup$ – rschwieb Mar 25 '13 at 13:51
  • $\begingroup$ Ah, makes much more sense now! Thanks! $\endgroup$ – Nicky Mar 25 '13 at 14:27
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The symbol $\oplus$ means direct sum.

The direct sum of two abelian groups $G$ and $H$ is the abelian group on the set $G\times H$ (cartesian product) with the group operation given by $(g,h) + (g',h') = (g+g',h+h')$.

You may well have seen this group denoted $G\times H$ and indeed, as long as the number of terms is finite, the direct sum and direct product of abelian groups are isomorphic.

More precisely, the direct sum is the coproduct in the category of abelian groups, while the direct product is the product.

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    $\begingroup$ And note that $\bigotimes$ does NOT mean direct product (but tensor product most of the time). $\endgroup$ – Nicky Hekster Mar 25 '13 at 14:31
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    $\begingroup$ To add to confusion, I've seen some authors (usually in physics texts) call the tensor product "the direct product". $\endgroup$ – rschwieb Mar 25 '13 at 18:44
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$\oplus$ denotes the direct sum.

http://en.wikipedia.org/wiki/Direct_sum

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