Since both notations are the same, why $\mathbb Z/2\mathbb Z\oplus \mathbb Z/2\mathbb Z$ is used instead of $\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z$ ?
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But it is not always used instead. – Tobias Kildetoft Nov 01 '16 at 12:28
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Wherever both apply, the group with $\oplus$ is "nicer" than the group with $\times$. So, one may be inclined to use $\oplus$ wherever it's possible to do so. – Ben Grossmann Nov 01 '16 at 12:33