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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2 - 2$\begingroup$ See math.stackexchange.com/questions/39895/… $\endgroup$lhf– lhf2021-06-23 10:43:00 +00:00Commented Jun 23, 2021 at 10:43
- 2$\begingroup$ 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). $\endgroup$Arsenaler– Arsenaler2021-06-23 10:46:59 +00:00Commented Jun 23, 2021 at 10:46
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4 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.
- $\begingroup$ 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. $\endgroup$Christoph– Christoph2021-06-23 10:54:30 +00:00Commented Jun 23, 2021 at 10:54
- 1$\begingroup$ @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 $\endgroup$Alan– Alan2021-06-23 10:58:08 +00:00Commented Jun 23, 2021 at 10:58
- $\begingroup$ I see, that is a very common "abuse" of notation indeed. $\endgroup$Christoph– Christoph2021-06-23 11:00:43 +00:00Commented Jun 23, 2021 at 11:00
- $\begingroup$ @Christoph Oh, yes. I'm just very pedantic :) $\endgroup$Alan– Alan2021-06-23 11:01:37 +00:00Commented Jun 23, 2021 at 11:01