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Are $\mathbb{Z}^2$ and $\mathbb{Z}\oplus \mathbb{Z}$ the same thing or are they different? I keep seeing both notations in a lot of mathematical literature, and I know elements in both are of the form $(a,b)$ where $a,b\in \mathbb{Z}$.

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    See https://math.stackexchange.com/questions/39895/the-direct-sum-oplus-versus-the-cartesian-product-times – lhf Jun 23 '21 at 10:43
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    They are the same. $\mathbb{Z}\times \mathbb{Z}$ is the direct product of $\mathbb{Z}$ with itself, $\mathbb{Z}\oplus\mathbb{Z}$ is the direct sum of $\mathbb{Z}$ and itself, and direct product and direct sum are the same when the situation involving is finite (they are different if the numbers of elements in the product are infinite). – Arsenaler Jun 23 '21 at 10:46

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Technically, $\mathbb{Z}^2=\mathbb{Z} \times \mathbb{Z}$ whereas $\mathbb{Z}\oplus \mathbb{Z}$ is the same set along with the associated group operation of pairwise addition inherited from the group $(\mathbb{Z},+)$. We often use the former by abuse of notation.

Alan
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  • There's no abuse of notation here. The category of abelian groups has products and coproducts, they just happen to be canonically isomorphic if the number of factors (here $2$) is finite. – Christoph Jun 23 '21 at 10:54
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    @Christoph The abuse of notation is associating $\mathbb{Z}$ which is a set with the group $(\mathbb{Z},+)$, not in the cross product then carrying the group operation – Alan Jun 23 '21 at 10:58
  • I see, that is a very common "abuse" of notation indeed. – Christoph Jun 23 '21 at 11:00
  • @Christoph Oh, yes. I'm just very pedantic :) – Alan Jun 23 '21 at 11:01